INTERNATIONAL GONGRESS 0 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z">|ClK[ℓ]| bounded by (2.5) for all ℓ ℓ\ellℓ; and infinitely many totally real degree n S n n S n nS_(n)n S_{n}nSn-fields K K KKK with C l K C l K Cl_(K)\mathrm{Cl}_{K}ClK containing an element of exact order ℓ ℓ\ellℓ and | C l K [ ] | C l K [ ℓ ] |Cl_(K)[ℓ]|\left|\mathrm{Cl}_{K}[\ell]\right||ClK[ℓ]| bounded by (2.5).) What happens when G G GGG does not have a unique minimal nontrivial normal subgroup? Here is an open question: in general, when N N NNN is a nontrivial normal subgroup of G G GGG (not necessarily unique or minimal), what is the true order of growth of (4.12) as X X → ∞ X rarr ooX \rightarrow \inftyX→∞ ? Questions about this "intersection multiplicity" are gathered in [62].
Third, increased attention has turned to bounding â„“ â„“\ellâ„“-torsion in class groups for all fields in special families specified by the Galois group: that is, proving property C F , ( Δ ) C F , â„“ ( Δ ) C_(F,â„“)(Delta)\mathbf{C}_{\mathscr{F}, \ell}(\Delta)CF,â„“(Δ) for some Δ < 1 / 2 Δ < 1 / 2 Delta < 1//2\Delta<1 / 2Δ<1/2. First, Klüners and Wang have proved C F , p ( 0 ) C F , p ( 0 ) C_(F,p)(0)\mathbf{C}_{\mathscr{F}, p}(0)CF,p(0) for the family F p r ( G ; X ) F p r ( G ; X ) F_(p^(r))(G;X)\mathscr{F}_{p^{r}}(G ; X)Fpr(G;X) for any p p ppp-group G G GGG; this generalizes the application of genus theory to prove C 2 , 2 ( 0 ) C 2 , 2 ( 0 ) C_(2,2)(0)\mathbf{C}_{2,2}(0)C2,2(0) [54].
Second, let G = ( Z / p Z ) r G = ( Z / p Z ) r G=(Z//pZ)^(r)G=(\mathbb{Z} / p \mathbb{Z})^{r}G=(Z/pZ)r be an elementary abelian group of rank r 2 r ≥ 2 r >= 2r \geq 2r≥2, with p p ppp prime. Wang has shown that for every â„“ â„“\ellâ„“, within the family of Galois G G GGG-fields K / Q K / Q K//QK / \mathbb{Q}K/Q, property C F , ( 1 / 2 δ ( , p ) ) C F , â„“ ( 1 / 2 − δ ( â„“ , p ) ) C_(F,â„“)(1//2-delta(â„“,p))\mathbf{C}_{\mathscr{F}, \ell}(1 / 2-\delta(\ell, p))CF,â„“(1/2−δ(â„“,p)) holds for some δ ( , p ) > 0 δ ( â„“ , p ) > 0 delta(â„“,p) > 0\delta(\ell, p)>0δ(â„“,p)>0 [91]. Since the savings δ ( , p ) δ ( â„“ , p ) delta(â„“,p)\delta(\ell, p)δ(â„“,p) is independent of the rank, for r r rrr sufficiently large this is better than C F , ( Δ G R H ) C F , â„“ Δ G R H C_(F,â„“)(Delta_(GRH))\mathbf{C}_{\mathscr{F}, \ell}\left(\Delta_{\mathrm{GRH}}\right)CF,â„“(ΔGRH). The method of proof plays off the interaction of three facts arising from the precise structure of G G GGG : first, | C l K [ ] | C l K [ â„“ ] |Cl_(K)[â„“]|\left|\mathrm{Cl}_{K}[\ell]\right||ClK[â„“]| factors as a product of | C l F [ ] | C l F [ â„“ ] |Cl_(F)[â„“]|\left|\mathrm{Cl}_{F}[\ell]\right||ClF[â„“]| where F F FFF varies over the p r 1 ≈ p r − 1 ~~p^(r-1)\approx p^{r-1}≈pr−1 many degree p p ppp subfields of K K KKK, so it suffices to bound one of these factors nontrivially. Second, any rational prime splits completely in p r 2 ≈ p r − 2 ~~p^(r-2)\approx p^{r-2}≈pr−2 of these subfields, so at least one subfield has a positive proportion of primes splitting completely in it. Third, the sizes of the discriminants of the subfields can be played against each other, so that known prime-counting results (which may a priori seem to count primes that are "too large") suffice for the application of the Ellenberg-Venkatesh criterion. This is an interesting counterpoint to the methods described earlier. In another direction, Wang has developed the notion of a forcing extension; certain nilpotent groups can be built from elementary p p ppp-groups via forcing extensions. If G G ′ G^(')G^{\prime}G′ is constructed from G G GGG by a forcing extension, then C F , ( Δ ) C F ′ , â„“ Δ ′ C_(F^('),â„“)(Delta^('))\mathbf{C}_{\mathscr{F}^{\prime}, \ell}\left(\Delta^{\prime}\right)CF′,â„“(Δ′) can be deduced from C F , ( Δ ) C F , â„“ ( Δ ) C_(F,â„“)(Delta)\mathbf{C}_{\mathscr{F}, \ell}(\Delta)CF,â„“(Δ), for some Δ , Δ < 1 / 2 Δ , Δ ′ < 1 / 2 Delta,Delta^(') < 1//2\Delta, \Delta^{\prime}<1 / 2Δ,Δ′<1/2, where F F F\mathscr{F}F is the family of G G GGG-extensions and F F ′ F^(')\mathscr{F}^{\prime}F′ is the family of G G ′ G^(')G^{\prime}G′-extensions [89].
All of the results mentioned in this section (except where genus theory suffices) directly apply or build on the Ellenberg-Venkatesh criterion. Can this criterion be strengthened? Ellenberg has suggested some possible improvements in [32]. In particular, let η ( K ) := inf { H K ( α ) : K = Q ( α ) } η ( K ) := inf H K ( α ) : K = Q ( α ) eta(K):=i n f{H_(K)(alpha):K=Q(alpha)}\eta(K):=\inf \left\{H_{K}(\alpha): K=\mathbb{Q}(\alpha)\right\}η(K):=inf{HK(α):K=Q(α)} denote the minimum (relative) multiplicative Weil height of a generating element of K K KKK. Roughly speaking, Ellenberg notes the criterion (2.4) can actually allow prime ideals with norms as large as η ( K ) 1 / η ( K ) 1 / â„“ eta(K)^(1//â„“)\eta(K)^{1 / \ell}η(K)1/â„“. The restriction to norms < D K 1 2 ( n 1 ) < D K 1 2 â„“ ( n − 1 ) < D_(K)^((1)/(2â„“(n-1)))<D_{K}^{\frac{1}{2 \ell(n-1)}}<DK12â„“(n−1) in (2.4) was made since the lower bound η ( K ) D K 1 2 ( n 1 ) η ( K ) ≥ D K 1 2 ( n − 1 ) eta(K) >= D_(K)^((1)/(2(n-1)))\eta(K) \geq D_{K}^{\frac{1}{2(n-1)}}η(K)≥DK12(n−1) holds for all fields [80]. Widmer, also with Frei, has shown that η ( K ) η ( K ) eta(K)\eta(K)η(K) can be enlarged for almost all fields in certain families, leading to improved upper bounds for â„“ â„“\ellâ„“-torsion in those fields [41,95]. That is, they improve the very notion of the "GRH-bound" (2.5), and show that the parameter we have called Δ GRH Δ GRH  Delta_("GRH ")\Delta_{\text {GRH }}ΔGRH  can actually be taken smaller for some fields. Their work raises interesting open questions: what upper and lower bounds hold for η ( K ) η ( K ) eta(K)\eta(K)η(K), for all (or almost all) fields in a family? Ruppert [74] has conjectured uniform upper bounds η ( K ) D K 1 / 2 η ( K ) ≤ D K 1 / 2 eta(K) <= D_(K)^(1//2)\eta(K) \leq D_{K}^{1 / 2}η(K)≤DK1/2 (now proved for almost all fields in some families by [72]). If this is true, the Ellenberg-Venkatesh criterion would hit a barrier, for most fields, with a result like | C l K [ ] | D K 1 / 2 1 / 2 + ε C l K [ â„“ ] ≪ D K 1 / 2 − 1 / 2 â„“ + ε |Cl_(K)[â„“]|≪D_(K)^(1//2-1//2â„“+epsi)\left|\mathrm{Cl}_{K}[\ell]\right| \ll D_{K}^{1 / 2-1 / 2 \ell+\varepsilon}|ClK[â„“]|≪DK1/2−1/2â„“+ε for any degree n n nnn, still far from the â„“ â„“\ellâ„“-torsion Conjecture. It would be very interesting to find a new, different criterion.

5. WHY DO WE EXPECT THE â„“ â„“\ellâ„“-TORSION CONJECTURE TO BE TRUE?

Recall that the â„“ â„“\ellâ„“-torsion Conjecture 2.1 is still known only in the case stemming from Gauss's work, namely for n = 2 , = 2 n = 2 , â„“ = 2 n=2,â„“=2n=2, \ell=2n=2,â„“=2. It is a good idea to affirm why we believe the â„“ â„“\ellâ„“-torsion Conjecture should be true. We will consider this from three perspectives.

5.1. From the perspective of the Cohen-Lenstra-Martinet heuristics

So far, when we have mentioned a result for almost all fields in a family, we have not focused on the size of a potential exceptional set, other than showing it is smaller than the size of the full family. But to understand the â„“ â„“\ellâ„“-torsion Conjecture, we must quantify a potential exceptional set, and show that for all sufficiently large discriminants, it is empty.
Let us abstract this, for a family F 0 ( X ) F 0 ( X ) F_(0)(X)\mathscr{F}_{0}(X)F0(X) of fields K K KKK with D K D K D_(K)D_{K}DK in a dyadic range ( X / 2 , X ] ( X / 2 , X ] (X//2,X](X / 2, X](X/2,X], from which more general results can easily be deduced by summing over log X ≪ log ⁡ X ≪log X\ll \log X≪log⁡X dyadic ranges. Suppose f : F 0 ( X ) N f : F 0 ( X ) → N f:F_(0)(X)rarrNf: \mathscr{F}_{0}(X) \rightarrow \mathbb{N}f:F0(X)→N is a function with f ( K ) D K a f ( K ) ≤ D K a f(K) <= D_(K)^(a)f(K) \leq D_{K}^{a}f(K)≤DKa for all K K KKK. Suppose that for some Δ < a Δ < a Delta < a\Delta<aΔ<a we can improve this to f ( K ) D K Δ f ( K ) ≤ D K Δ f(K) <= D_(K)^(Delta)f(K) \leq D_{K}^{\Delta}f(K)≤DKΔ for all K K KKK outside of some exceptional set E 0 Δ ( X ) F 0 ( X ) E 0 Δ ( X ) ⊂ F 0 ( X ) E_(0)^(Delta)(X)subF_(0)(X)E_{0}^{\Delta}(X) \subset \mathscr{F}_{0}(X)E0Δ(X)⊂F0(X). Then
K F 0 ( X ) f ( K ) = K F 0 ( X ) E 0 Δ ( X ) f ( K ) + K E 0 Δ ( X ) f ( K ) | F 0 ( X ) | X Δ + | E 0 Δ ( X ) | X a ∑ K ∈ F 0 ( X )   f ( K ) = ∑ K ∈ F 0 ( X ) ∖ E 0 Δ ( X )   f ( K ) + ∑ K ∈ E 0 Δ ( X )   f ( K ) ≤ F 0 ( X ) X Δ + E 0 Δ ( X ) X a sum_(K inF_(0)(X))f(K)=sum_(K inF_(0)(X)\\E_(0)^(Delta)(X))f(K)+sum_(K inE_(0)^(Delta)(X))f(K) <= |F_(0)(X)|X^(Delta)+|E_(0)^(Delta)(X)|X^(a)\sum_{K \in \mathscr{F}_{0}(X)} f(K)=\sum_{K \in \mathscr{F}_{0}(X) \backslash E_{0}^{\Delta}(X)} f(K)+\sum_{K \in E_{0}^{\Delta}(X)} f(K) \leq\left|\mathscr{F}_{0}(X)\right| X^{\Delta}+\left|E_{0}^{\Delta}(X)\right| X^{a}∑K∈F0(X)f(K)=∑K∈F0(X)∖E0Δ(X)f(K)+∑K∈E0Δ(X)f(K)≤|F0(X)|XΔ+|E0Δ(X)|Xa.
As long as | E 0 Δ ( X ) | | F 0 ( X ) | X ( a Δ ) E 0 Δ ( X ) ≪ F 0 ( X ) X − ( a − Δ ) |E_(0)^(Delta)(X)|≪|F_(0)(X)|X^(-(a-Delta))\left|E_{0}^{\Delta}(X)\right| \ll\left|\mathscr{F}_{0}(X)\right| X^{-(a-\Delta)}|E0Δ(X)|≪|F0(X)|X−(a−Δ), this shows that f ( K ) X Δ f ( K ) ≪ X Δ f(K)≪X^(Delta)f(K) \ll X^{\Delta}f(K)≪XΔ on average. On the other hand, suppose we know K F 0 ( X ) f ( K ) X b ∑ K ∈ F 0 ( X )   f ( K ) ≤ X b sum_(K inF_(0)(X))f(K) <= X^(b)\sum_{K \in \mathscr{F}_{0}(X)} f(K) \leq X^{b}∑K∈F0(X)f(K)≤Xb. Then a potential set of exceptions E 0 Δ ( X ) = { K F 0 ( X ) : f ( K ) > D K Δ } E 0 Δ ( X ) = K ∈ F 0 ( X ) : f ( K ) > D K Δ E_(0)^(Delta)(X)={K inF_(0)(X):f(K) > D_(K)^(Delta)}E_{0}^{\Delta}(X)=\left\{K \in \mathscr{F}_{0}(X): f(K)>D_{K}^{\Delta}\right\}E0Δ(X)={K∈F0(X):f(K)>DKΔ} can be controlled by
(5.2) X Δ | E 0 Δ ( X ) | K E 0 Δ ( X ) f ( K ) K F 0 ( X ) f ( K ) X b (5.2) X Δ E 0 Δ ( X ) ≪ ∑ K ∈ E 0 Δ ( X )   f ( K ) ≤ ∑ K ∈ F 0 ( X )   f ( K ) ≤ X b {:(5.2)X^(Delta)|E_(0)^(Delta)(X)|≪sum_(K inE_(0)^(Delta)(X))f(K) <= sum_(K inF_(0)(X))f(K) <= X^(b):}\begin{equation*} X^{\Delta}\left|E_{0}^{\Delta}(X)\right| \ll \sum_{K \in E_{0}^{\Delta}(X)} f(K) \leq \sum_{K \in \mathscr{F}_{0}(X)} f(K) \leq X^{b} \tag{5.2} \end{equation*}(5.2)XΔ|E0Δ(X)|≪∑K∈E0Δ(X)f(K)≤∑K∈F0(X)f(K)≤Xb
Thus | E 0 Δ ( X ) | X b Δ E 0 Δ ( X ) ≪ X b − Δ |E_(0)^(Delta)(X)|≪X^(b-Delta)\left|E_{0}^{\Delta}(X)\right| \ll X^{b-\Delta}|E0Δ(X)|≪Xb−Δ, and exceptional fields are density zero in F 0 ( X ) F 0 ( X ) F_(0)(X)\mathscr{F}_{0}(X)F0(X), provided X b Δ = o ( | F 0 ( X ) | ) X b − Δ = o F 0 ( X ) X^(b-Delta)=o(|F_(0)(X)|)X^{b-\Delta}=o\left(\left|\mathscr{F}_{0}(X)\right|\right)Xb−Δ=o(|F0(X)|). That is, a nontrivial upper bound on â„“ â„“\ellâ„“-torsion for "almost all" fields in a family F F F\mathscr{F}F is essentially equivalent to the same upper bound "on average."
To verify the â„“ â„“\ellâ„“-torsion Conjecture, we wish to show a "pointwise" bound: for every ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0, there exists D ε D ε D_(epsi)D_{\varepsilon}Dε such that when D K D ε D K ≥ D ε D_(K) >= D_(epsi)D_{K} \geq D_{\varepsilon}DK≥Dε, there are no exceptions to the bound | C l K [ ] | D K ε C l K [ â„“ ] ≤ D K ε |Cl_(K)[â„“]| <= D_(K)^(epsi)\left|\mathrm{Cl}_{K}[\ell]\right| \leq D_{K}^{\varepsilon}|ClK[â„“]|≤DKε. The key is to consider not averages but arbitrarily high k k kkk th moments. In the general setting above, suppose that we know K F 0 ( X ) f ( K ) k X b ∑ K ∈ F 0 ( X )   f ( K ) k ≤ X b sum_(K inF_(0)(X))f(K)^(k) <= X^(b)\sum_{K \in \mathscr{F}_{0}(X)} f(K)^{k} \leq X^{b}∑K∈F0(X)f(K)k≤Xb, for a real number k 1 k ≥ 1 k >= 1k \geq 1k≥1. Then for any fixed Δ > 0 Δ > 0 Delta > 0\Delta>0Δ>0, adapting the argument (5.2) shows that | E 0 Δ ( X ) | X b k Δ E 0 Δ ( X ) ≪ X b − k Δ |E_(0)^(Delta)(X)|≪X^(b-k Delta)\left|E_{0}^{\Delta}(X)\right| \ll X^{b-k \Delta}|E0Δ(X)|≪Xb−kΔ. If the k k kkk th moment is uniformly bounded by X b X b X^(b)X^{b}Xb for a sequence of k k → ∞ k rarr ook \rightarrow \inftyk→∞, then for each Δ > 0 Δ > 0 Delta > 0\Delta>0Δ>0, we can take k k kkk sufficiently large to conclude that the set of exceptions is empty.
This perspective has been applied by Pierce, Turnage-Butterbaugh, and Wood in [73] to prove that the â„“ â„“\ellâ„“-torsion Conjecture holds for all fields in a family F ( X ) F ( X ) F(X)\mathscr{F}(X)F(X) if there is a real number α 1 α ≥ 1 alpha >= 1\alpha \geq 1α≥1 such that for a sequence of arbitrarily large k k kkk,
(5.3) K F ( X ) | C l K [ ] | k n , , k , α | F ( X ) | α , for all X 1 (5.3) ∑ K ∈ F ( X )   C l K [ â„“ ] k ≪ n , â„“ , k , α | F ( X ) | α ,  for all  X ≥ 1 {:(5.3)sum_(K inF(X))|Cl_(K)[â„“]|^(k)≪_(n,â„“,k,alpha)|F(X)|^(alpha)","quad" for all "X >= 1:}\begin{equation*} \sum_{K \in \mathscr{F}(X)}\left|\mathrm{Cl}_{K}[\ell]\right|^{k} \ll_{n, \ell, k, \alpha}|\mathscr{F}(X)|^{\alpha}, \quad \text { for all } X \geq 1 \tag{5.3} \end{equation*}(5.3)∑K∈F(X)|ClK[â„“]|k≪n,â„“,k,α|F(X)|α, for all X≥1
The Cohen-Lenstra-Martinet heuristics predict that (5.3) holds, in the form of an even stronger asymptotic with α = 1 α = 1 alpha=1\alpha=1α=1, for all integers k 1 k ≥ 1 k >= 1k \geq 1k≥1, for families of Galois G G GGG-extensions, at least for all primes | G | â„“ ∤ | G | ℓ∤|G|\ell \nmid|G|ℓ∤|G|. The appropriate moment formulation can be found in [21] for degree 2 fields and in [92] for higher degrees, building on [22]. This confirms that the â„“ â„“\ellâ„“-torsion Conjecture follows from another well-known set of conjectures.
The Cohen-Lenstra-Martinet heuristics are a subject of intense interest and much recent activity. Here are some spectacular successes most closely related to our topic. Dav-
enport and Heilbronn [28] have proved
(5.4) deg ( K ) = 2 0 < D K X | C l K [ 3 ] | ( 2 3 ζ ( 2 ) + 1 ζ ( 2 ) ) X (5.4) ∑ deg ⁡ ( K ) = 2 0 < D K ≤ X   C l K [ 3 ] ∼ 2 3 ζ ( 2 ) + 1 ζ ( 2 ) X {:(5.4)sum_({:[deg(K)=2],[0 < D_(K) <= X]:})|Cl_(K)[3]|∼((2)/(3zeta(2))+(1)/(zeta(2)))X:}\begin{equation*} \sum_{\substack{\operatorname{deg}(K)=2 \\ 0<D_{K} \leq X}}\left|\mathrm{Cl}_{K}[3]\right| \sim\left(\frac{2}{3 \zeta(2)}+\frac{1}{\zeta(2)}\right) X \tag{5.4} \end{equation*}(5.4)∑deg⁡(K)=20<DK≤X|ClK[3]|∼(23ζ(2)+1ζ(2))X
second-order terms have been found in [5,11,85]. Bhargava [7] has proved
(5.5) deg ( K ) = 3 0 < D K X | C l K [ 2 ] | ( 5 48 ζ ( 3 ) + 3 8 ζ ( 3 ) ) X (5.5) ∑ deg ⁡ ( K ) = 3 0 < D K ≤ X   C l K [ 2 ] ∼ 5 48 ζ ( 3 ) + 3 8 ζ ( 3 ) X {:(5.5)sum_({:[deg(K)=3],[0 < D_(K) <= X]:})|Cl_(K)[2]|∼((5)/(48 zeta(3))+(3)/(8zeta(3)))X:}\begin{equation*} \sum_{\substack{\operatorname{deg}(K)=3 \\ 0<D_{K} \leq X}}\left|\mathrm{Cl}_{K}[2]\right| \sim\left(\frac{5}{48 \zeta(3)}+\frac{3}{8 \zeta(3)}\right) X \tag{5.5} \end{equation*}(5.5)∑deg⁡(K)=30<DK≤X|ClK[2]|∼(548ζ(3)+38ζ(3))X
in which each isomorphism class of fields is counted once. Very recently, [63] obtained analogues of (5.4) for averages over F 2 m ( G ; X ) F 2 m ( G ; X ) F_(2^(m))(G;X)\mathscr{F}_{2^{m}}(G ; X)F2m(G;X) for any permutation group G S 2 m G ⊂ S 2 m G subS_(2^(m))G \subset S_{2^{m}}G⊂S2m that is a transitive permutation 2-group containing a transposition. See also the work of Smith on the distribution of 2 k 2 k 2^(k)2^{k}2k-class groups in imaginary quadratic fields [81]; Koymans and Pagano on k ℓ k ℓ^(k)\ell^{k}ℓk-class groups of degree ℓ ℓ\ellℓ cyclic fields [59]; Klys on moments of p p ppp-torsion in cyclic degree p p ppp fields (conditional on GRH for p 5 p ≥ 5 p >= 5p \geq 5p≥5 ) [55]; Milovic and Koymans on 16-rank in quadratic fields [57,58]; Bhargava and Varma [13,14] elaborating on (5.4) and (5.5).
The perspective of moments (5.3) provides a strong motivation to prove the k k kkk th moment bounds for ℓ ℓ\ellℓ-torsion. Fouvry and Klüners have proved an asymptotic for the k k kkk th moments related to 4-torsion when K K KKK is quadratic, for all integers k 1 k ≥ 1 k >= 1k \geq 1k≥1 [38]. Heath-Brown and Pierce have proved nontrivial bounds for the k k kkk th moments of ℓ ℓ\ellℓ-torsion for imaginary quadratic fields, for all odd primes ℓ ℓ\ellℓ [46]. For example, they establish second moment bounds
(5.6) K = Q ( ± D ) D X | C l K [ 3 ] | 2 X 23 / 18 , K = Q ( D ) D X | C l K [ ] | 2 X 2 3 + 1 , 5 prime (5.6) ∑ K = Q ( ± D ) D ≤ X   C l K [ 3 ] 2 ≪ X 23 / 18 , ∑ K = Q ( − D ) D ≤ X   C l K [ â„“ ] 2 ≪ X 2 − 3 â„“ + 1 , â„“ ≥ 5  prime  {:(5.6)sum_({:[K=Q(sqrt(+-D))],[D <= X]:})|Cl_(K)[3]|^(2)≪X^(23//18)","quadsum_({:[K=Q(sqrt(-D))],[D <= X]:})|Cl_(K)[â„“]|^(2)≪X^(2-(3)/(â„“+1))","quadâ„“ >= 5" prime ":}\begin{equation*} \sum_{\substack{K=\mathbb{Q}(\sqrt{ \pm D}) \\ D \leq X}}\left|\mathrm{Cl}_{K}[3]\right|^{2} \ll X^{23 / 18}, \quad \sum_{\substack{K=\mathbb{Q}(\sqrt{-D}) \\ D \leq X}}\left|\mathrm{Cl}_{K}[\ell]\right|^{2} \ll X^{2-\frac{3}{\ell+1}}, \quad \ell \geq 5 \text { prime } \tag{5.6} \end{equation*}(5.6)∑K=Q(±D)D≤X|ClK[3]|2≪X23/18,∑K=Q(−D)D≤X|ClK[â„“]|2≪X2−3â„“+1,ℓ≥5 prime 
as well as results for the k k kkk th moments for all k 1 k ≥ 1 k >= 1k \geq 1k≥1. In general, proving tighter control on the size of an exceptional family E 0 Δ ( X ) E 0 Δ ( X ) E_(0)^(Delta)(X)E_{0}^{\Delta}(X)E0Δ(X) can be used to deduce a better moment bound for | C l K [ ] | C l K [ â„“ ] |Cl_(K)[â„“]|\left|\mathrm{Cl}_{K}[\ell]\right||ClK[â„“]|, similar to (5.1). This has recently been exploited by Frei and Widmer, in combination with refinements of the Ellenberg-Venkatesh criterion, to improve moment bounds on â„“ â„“\ellâ„“-torsion for the families of fields studied in [72] (if â„“ â„“\ellâ„“ is sufficiently large); see [41].
Let us mention a connection to elliptic curves; this was after all the setting in which Brumer and Silverman initially posed the â„“ â„“\ellâ„“-torsion Conjecture. Let E ( q ) E ( q ) E(q)E(q)E(q) denote the number of isomorphism classes of elliptic curves over Q Q Q\mathbb{Q}Q with conductor q q qqq. Brumer and Silverman have conjectured that E ( q ) ε q ε E ( q ) ≪ ε q ε E(q)≪_(epsi)q^(epsi)E(q) \ll_{\varepsilon} q^{\varepsilon}E(q)≪εqε for every q 1 , ε > 0 q ≥ 1 , ε > 0 q >= 1,epsi > 0q \geq 1, \varepsilon>0q≥1,ε>0 [17]. Conditionally, this follows from GRH combined with a weak form of the Birch-Swinnerton-Dyer conjecture. They also showed this follows from the 3-torsion Conjecture for quadratic fields, by proving
(5.7) E ( q ) ε q ε max 1 D 1728 q | C l Q ( ± D ) [ 3 ] | , for all ε > 0 (5.7) E ( q ) ≪ ε q ε max 1 ≤ D ≤ 1728 q   C l Q ( ± D ) [ 3 ] ,  for all  ε > 0 {:(5.7)E(q)≪_(epsi)q^(epsi)max_(1 <= D <= 1728 q)|Cl_(Q(sqrt(+-D)))[3]|","quad" for all "epsi > 0:}\begin{equation*} E(q) \ll_{\varepsilon} q^{\varepsilon} \max _{1 \leq D \leq 1728 q}\left|\mathrm{Cl}_{\mathbb{Q}(\sqrt{ \pm D})}[3]\right|, \quad \text { for all } \varepsilon>0 \tag{5.7} \end{equation*}(5.7)E(q)≪εqεmax1≤D≤1728q|ClQ(±D)[3]|, for all ε>0
Duke and Kowalski have combined this with the celebrated asymptotic (5.4) to bound 1 q Q E ( q ) Q 1 + ε ∑ 1 ≤ q ≤ Q   E ( q ) ≪ Q 1 + ε sum_(1 <= q <= Q)E(q)≪Q^(1+epsi)\sum_{1 \leq q \leq Q} E(q) \ll Q^{1+\varepsilon}∑1≤q≤QE(q)≪Q1+ε for every ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 [30]. (See also [39] for ordering by discriminant.) Pierce, Turnage-Butterbaugh, and Wood have recently proved that for all k 1 k ≥ 1 k >= 1k \geq 1k≥1, the k k kkk th moment of 3-torsion in quadratic fields dominates the γ k γ k gamma k\gamma kγk th moment of E ( q ) E ( q ) E(q)E(q)E(q), for a numerical constant γ 1.9745 γ ≈ 1.9745 … gamma~~1.9745 dots\gamma \approx 1.9745 \ldotsγ≈1.9745… coming from [48], which sharpened the relation (5.7). Thus
new moment bounds for E ( q ) E ( q ) E(q)E(q)E(q) can be obtained from (5.6), for example. Here is an open problem: prove that 1 q Q E ( q ) = o ( Q ) ∑ 1 ≤ q ≤ Q   E ( q ) = o ( Q ) sum_(1 <= q <= Q)E(q)=o(Q)\sum_{1 \leq q \leq Q} E(q)=o(Q)∑1≤q≤QE(q)=o(Q). This would show for the first time that integers that are the conductor of an elliptic curve have density zero in Z Z Z\mathbb{Z}Z. In fact, it is conjectured by Watkins that this average is asymptotic to c Q 5 / 6 c Q 5 / 6 cQ^(5//6)c Q^{5 / 6}cQ5/6 for a certain constant c c ccc [94] (building on an analogous conjecture by Brumer-McGuinness for ordering by discriminant [16]).
To conclude, in this section we saw that the truth of the â„“ â„“\ellâ„“-torsion Conjecture is implied by the truth of the well-known Cohen-Lenstra-Martinet heuristics on the distribution of class groups.

5.2. From the perspective of counting number fields of fixed discriminant

Let K / Q K / Q K//QK / \mathbb{Q}K/Q be a degree n n nnn extension. The Hilbert class field H K H K H_(K)H_{K}HK is the maximal abelian unramified extension of K K KKK, and C l K C l K Cl_(K)\mathrm{Cl}_{K}ClK is isomorphic to Gal ( H K / K ) Gal ⁡ H K / K Gal(H_(K)//K)\operatorname{Gal}\left(H_{K} / K\right)Gal⁡(HK/K). A second way to motivate the ℓ ℓ\ellℓ-torsion Conjecture is to count intermediate fields between K K KKK and H K H K H_(K)H_{K}HK.
Here is an argument recorded by Pierce, Turnage-Butterbaugh, and Wood in [73]. Fix a prime ℓ ℓ\ellℓ and write C l K C l K Cl_(K)\mathrm{Cl}_{K}ClK additively, so that C l K [ ] C l K / C l K C l K [ ℓ ] ≃ C l K / ℓ C l K Cl_(K)[ℓ]≃Cl_(K)//ℓCl_(K)\mathrm{Cl}_{K}[\ell] \simeq \mathrm{Cl}_{K} / \ell \mathrm{Cl}_{K}ClK[ℓ]≃ClK/ℓClK. Now define the fixed field L = H K C l K L = H K ℓ C l K L=H_(K)^(ℓCl_(K))L=H_{K}^{\ell \mathrm{Cl}_{K}}L=HKℓClK lying between K K KKK and H K H K H_(K)H_{K}HK, so Gal ( L / K ) C l K [ ] Gal ⁡ ( L / K ) ≃ C l K [ ℓ ] Gal(L//K)≃Cl_(K)[ℓ]\operatorname{Gal}(L / K) \simeq \mathrm{Cl}_{K}[\ell]Gal⁡(L/K)≃ClK[ℓ]. Each surjection C l K [ ] Z / Z C l K [ ℓ ] → Z / ℓ Z Cl_(K)[ℓ]rarrZ//ℓZ\mathrm{Cl}_{K}[\ell] \rightarrow \mathbb{Z} / \ell \mathbb{Z}ClK[ℓ]→Z/ℓZ generates an intermediate field M M MMM, with K M L K ⊂ M ⊂ L K sub M sub LK \subset M \subset LK⊂M⊂L and deg ( M / Q ) = n deg ⁡ ( M / Q ) = n ℓ deg(M//Q)=nℓ\operatorname{deg}(M / \mathbb{Q})=n \elldeg⁡(M/Q)=nℓ. If | C l K [ ] | = r C l K [ ℓ ] = ℓ r |Cl_(K)[ℓ]|=ℓ^(r)\left|\mathrm{Cl}_{K}[\ell]\right|=\ell^{r}|ClK[ℓ]|=ℓr, say, this produces r 1 ≈ ℓ r − 1 ~~ℓ^(r-1)\approx \ell^{r-1}≈ℓr−1 such fields M M MMM. The crucial point is that since H K H K H_(K)H_{K}HK is an unramified extension, all these fields satisfy a rigid discriminant identity D M = D K D M = D K ℓ D_(M)=D_(K)^(ℓ)D_{M}=D_{K}^{\ell}DM=DKℓ. Consequently, if we can count how many number fields of degree n n ℓ nℓn \ellnℓ can share the same fixed discriminant, then we can bound ℓ ℓ\ellℓ-torsion in C l K C l K Cl_(K)\mathrm{Cl}_{K}ClK. (We have seen this problem before.) We formalize the problem of counting number fields of fixed discriminant as follows:
Property D n ( Δ ) D n ( Δ ) D_(n)(Delta)\mathbf{D}_{n}(\Delta)Dn(Δ). Fix a degree n 2 n ≥ 2 n >= 2n \geq 2n≥2. Property D n ( Δ ) D n ( Δ ) D_(n)(Delta)\mathbf{D}_{n}(\Delta)Dn(Δ) holds if for every ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 and for every fixed integer D > 1 D > 1 D > 1D>1D>1, at most n , ε D Δ + ε ≪ n , ε D Δ + ε ≪_(n,epsi)D^(Delta+epsi)\ll_{n, \varepsilon} D^{\Delta+\varepsilon}≪n,εDΔ+ε fields K / Q K / Q K//QK / \mathbb{Q}K/Q of degree n n nnn have D K = D D K = D D_(K)=DD_{K}=DDK=D.
The strategy sketched above ultimately proves that property D n ( Δ ) D n â„“ ( Δ ) D_(nâ„“)(Delta)\mathbf{D}_{n \ell}(\Delta)Dnâ„“(Δ) implies C n , ( Δ ) C n , â„“ ( â„“ Δ ) C_(n,â„“)(â„“Delta)\mathbf{C}_{n, \ell}(\ell \Delta)Cn,â„“(ℓΔ). This leads inevitably to the question: is property D n ( 0 ) D n â„“ ( 0 ) D_(nâ„“)(0)\mathbf{D}_{n \ell}(0)Dnâ„“(0) true? Here is a conjecture:
Conjecture 5.1 (Discriminant multiplicity conjecture). For each n 2 n ≥ 2 n >= 2n \geq 2n≥2, for every ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0, and for every integer D > 1 D > 1 D > 1D>1D>1, at most n , ε D ε ≪ n , ε D ε ≪_(n,epsi)D^(epsi)\ll_{n, \varepsilon} D^{\varepsilon}≪n,εDε fields K / Q K / Q K//QK / \mathbb{Q}K/Q of degree n n nnn have D K = D D K = D D_(K)=DD_{K}=DDK=D.
This conjecture has been recorded by Duke [29]. It implies the â„“ â„“\ellâ„“-torsion Conjecture, a link noted in [ 29 , 35 ] [ 29 , 35 ] [29,35][29,35][29,35] and quantified in [73]. Recall the conjecture (3.3) for counting all fields of degree n n nnn and discriminant D K X D K ≤ X D_(K) <= XD_{K} \leq XDK≤X. The Discriminant Multiplicity Conjecture for degree n n nnn would immediately imply N n ( X ) X 1 + ε N n ( X ) ≪ X 1 + ε N_(n)(X)≪X^(1+epsi)N_{n}(X) \ll X^{1+\varepsilon}Nn(X)≪X1+ε, which indicates its level of difficulty. Of course, in general, property D n ( Δ ) D n ( Δ ) D_(n)(Delta)\mathbf{D}_{n}(\Delta)Dn(Δ) implies N n ( X ) X 1 + Δ + ε N n ( X ) ≪ X 1 + Δ + ε N_(n)(X)≪X^(1+Delta+epsi)N_{n}(X) \ll X^{1+\Delta+\varepsilon}Nn(X)≪X1+Δ+ε for all ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0. (In terms of lower bounds, Ellenberg and Venkatesh have noted there can be D c / log log D ≫ D c / log ⁡ log ⁡ D ≫D^(c//log log D)\gg D^{c / \log \log D}≫Dc/log⁡log⁡D extensions K / Q K / Q K//QK / \mathbb{Q}K/Q with a fixed Galois group and fixed discriminant D D DDD [35].)
The Discriminant Multiplicity Conjecture posits that D n ( 0 ) D n ( 0 ) D_(n)(0)\mathbf{D}_{n}(0)Dn(0) holds for each n 2 n ≥ 2 n >= 2n \geq 2n≥2. This is true for n = 2 n = 2 n=2n=2n=2, but it is not known for any other degree. For degrees n = 3 , 4 , 5 n = 3 , 4 , 5 n=3,4,5n=3,4,5n=3,4,5, the best-known results currently are D 3 ( 1 / 3 ) D 3 ( 1 / 3 ) D_(3)(1//3)\mathbf{D}_{3}(1 / 3)D3(1/3) by [34]; D 4 ( 1 / 2 ) D 4 ( 1 / 2 ) D_(4)(1//2)\mathbf{D}_{4}(1 / 2)D4(1/2) as found in [52, 72, 73, 97]; D 5 ( 199 / 200 ) D 5 ( 199 / 200 ) D_(5)(199//200)\mathbf{D}_{5}(199 / 200)D5(199/200) as found in [33], building on [9,79]. Currently for n 6 n ≥ 6 n >= 6n \geq 6n≥6, the only result for
D n ( Δ ) D n ( Δ ) D_(n)(Delta)\mathbf{D}_{n}(\Delta)Dn(Δ) is a trivial consequence of counting fields of bounded discriminant, as in (3.4), so in particular Δ = c 0 ( log n ) 2 > 1 Δ = c 0 ( log ⁡ n ) 2 > 1 Delta=c_(0)(log n)^(2) > 1\Delta=c_{0}(\log n)^{2}>1Δ=c0(log⁡n)2>1 in those cases. It would be very interesting to improve the exponent known for D n ( Δ ) D n ( Δ ) D_(n)(Delta)\mathbf{D}_{n}(\Delta)Dn(Δ), for any fixed degree n 3 n ≥ 3 n >= 3n \geq 3n≥3.
As is the case for many of the problems surveyed in this paper, it can also be profitable to study the problem within a family F F F\mathscr{F}F of degree n n nnn extensions:
Property D F , n ( Δ ) D F , n ( Δ ) D_(F,n)(Delta)\mathbf{D}_{\mathscr{F}, n}(\Delta)DF,n(Δ). Fix a degree n 2 n ≥ 2 n >= 2n \geq 2n≥2. Property D F , n ( Δ ) D F , n ( Δ ) D_(F,n)(Delta)\mathbf{D}_{\mathscr{F}, n}(\Delta)DF,n(Δ) holds iffor every ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 and for every fixed integer D > 1 D > 1 D > 1D>1D>1, at most n , ε D Δ + ε ≪ n , ε D Δ + ε ≪_(n,epsi)D^(Delta+epsi)\ll_{n, \varepsilon} D^{\Delta+\varepsilon}≪n,εDΔ+ε fields K / Q K / Q K//QK / \mathbb{Q}K/Q in the family F F F\mathscr{F}F have D K = D D K = D D_(K)=DD_{K}=DDK=D.
This is the type of property Pierce, Turnage-Butterbaugh, and Wood used to control the collision problem, in the form (4.11) [72]. Property D F , n ( 0 ) D F , n ( 0 ) D_(F,n)(0)\mathbf{D}_{\mathscr{F}, n}(0)DF,n(0) has recently been proved by Klüners and Wang, for the family F = F n ( G ; X ) F = F n ( G ; X ) F=F_(n)(G;X)\mathscr{F}=\mathscr{F}_{n}(G ; X)F=Fn(G;X) of degree n G n G nGn GnG-extensions for any nilpotent group G G GGG. This was built from the truth of property C F , p ( 0 ) C F , p ( 0 ) C_(F,p)(0)\mathbf{C}_{\mathscr{F}, p}(0)CF,p(0) for F F F\mathscr{F}F being the family of Galois H H HHH-extensions for H H HHH a p p ppp-group, in [54]. There are many other cases where it is an interesting open problem to improve the known bound for Property D F , n ( Δ ) Property ⁡ D F , n ( Δ ) Property D_(F,n)(Delta)\operatorname{Property} \mathbf{D}_{\mathscr{F}, n}(\Delta)Property⁡DF,n(Δ).
To conclude, in this section we saw that the â„“ â„“\ellâ„“-torsion Conjecture follows from the Discriminant Multiplicity Conjecture. Now, recall that we saw in the context of bounding â„“ â„“\ellâ„“-torsion that uniform bounds for arbitrarily high moments can imply strong "pointwise" results for every field. Can the method of moments be used to approach the Discriminant Multiplicity Conjecture too? We turn to this idea next.

5.3. From the perspective of counting number fields of bounded discriminant

We come to a third motivation to believe the ℓ ℓ\ellℓ-torsion Conjecture. Recall the definition (3.1) of a family F n ( G ; X ) F n ( G ; X ) F_(n)(G;X)\mathscr{F}_{n}(G ; X)Fn(G;X) of degree n n nnn fields K / Q K / Q K//QK / \mathbb{Q}K/Q with Gal ( K ~ / Q ) Gal ⁡ ( K ~ / Q ) Gal( tilde(K)//Q)\operatorname{Gal}(\tilde{K} / \mathbb{Q})Gal⁡(K~/Q) isomorphic (as a permutation group) to a nontrivial transitive subgroup G S n G ⊆ S n G subeS_(n)G \subseteq S_{n}G⊆Sn. Each element g G g ∈ G g in Gg \in Gg∈G has an index defined by ind ( g ) = n o g ind ⁡ ( g ) = n − o g ind(g)=n-o_(g)\operatorname{ind}(g)=n-o_{g}ind⁡(g)=n−og, where o g o g o_(g)o_{g}og is the number of orbits of g g ggg when it acts on a set of n n nnn elements. Define a ( G ) a ( G ) a(G)a(G)a(G) according to a ( G ) 1 = min { ind ( g ) : 1 g G } a ( G ) − 1 = min { ind ⁡ ( g ) : 1 ≠ g ∈ G } a(G)^(-1)=min{ind(g):1!=g in G}a(G)^{-1}=\min \{\operatorname{ind}(g): 1 \neq g \in G\}a(G)−1=min{ind⁡(g):1≠g∈G}; we see that 1 n 1 a ( G ) 1 1 n − 1 ≤ a ( G ) ≤ 1 (1)/(n-1) <= a(G) <= 1\frac{1}{n-1} \leq a(G) \leq 11n−1≤a(G)≤1. Malle has made a well-known conjecture [65]:
Conjecture 5.2 (Malle). For each n 2 n ≥ 2 n >= 2n \geq 2n≥2, for each transitive subgroup G S n G ⊆ S n G subeS_(n)G \subseteq S_{n}G⊆Sn,
(5.8) | F n ( G ; X ) | G , ε X a ( G ) + ε , for all ε > 0 (5.8) F n ( G ; X ) ≪ G , ε X a ( G ) + ε ,  for all  ε > 0 {:(5.8)|F_(n)(G;X)|≪_(G,epsi)X^(a(G)+epsi)","quad" for all "epsi > 0:}\begin{equation*} \left|\mathscr{F}_{n}(G ; X)\right| \ll_{G, \varepsilon} X^{a(G)+\varepsilon}, \quad \text { for all } \varepsilon>0 \tag{5.8} \end{equation*}(5.8)|Fn(G;X)|≪G,εXa(G)+ε, for all ε>0
Also, | F n ( G ; X ) | G X a ( G ) F n ( G ; X ) ≫ G X a ( G ) |F_(n)(G;X)|≫_(G)X^(a(G))\left|\mathscr{F}_{n}(G ; X)\right| \gg_{G} X^{a(G)}|Fn(G;X)|≫GXa(G).
The full statement of this conjecture is an open problem. Its difficulty is indicated by the fact that it implies a positive solution to the inverse Galois problem for number fields. (A refinement in [66] specified a power of log X log ⁡ X log X\log Xlog⁡X in place of X ε X ε X^(epsi)X^{\varepsilon}Xε; counterexamples to this refinement have been found in [50], but the upper bound in (5.8) is expected to be true.)
Malle's Conjecture has been proved for abelian groups, with a strategy by Cohn [24], and asymptotic counts by Mäki [64], Wright [97]. For n = 3 , 4 , 5 n = 3 , 4 , 5 n=3,4,5n=3,4,5n=3,4,5, it is known for S n S n S_(n)S_{n}Sn by the asymptotic (3.3), and for D 4 D 4 D_(4)D_{4}D4 by Baily [3] (refined to an asymptotic in [20]). It is known for C 2 C 2 C_(2)C_{2}C2 ä¹™ H H HHH under mild conditions on H H HHH (in particular, for at least one group of order n n nnn for every even n n nnn ) by [53], and for S n × A S n × A S_(n)xx AS_{n} \times ASn×A with A A AAA an abelian group by [67, 90]. For prime
degree p D p p D p pD_(p)p D_{p}pDp-fields, upper and lower bounds are closely related to p p ppp-torsion in class groups of quadratic fields, and have been studied in [23,41,51].
For many groups, it is a difficult open problem to prove upper or lower bounds approaching Malle's prediction. In many results surveyed here, proving a lower bound for | F n ( G ; X ) | F n ( G ; X ) |F_(n)(G;X)|\left|\mathscr{F}_{n}(G ; X)\right||Fn(G;X)| has been an important step, to verify a result applies to "almost all" fields in a family. For many groups G G GGG, it is not even known that | F n ( G ; X ) | X β F n ( G ; X ) ≫ X β |F_(n)(G;X)|≫X^(beta)\left|\mathscr{F}_{n}(G ; X)\right| \gg X^{\beta}|Fn(G;X)|≫Xβ for some β > 0 β > 0 beta > 0\beta>0β>0 as X X → ∞ X rarr ooX \rightarrow \inftyX→∞. Here is a tool to prove such a result: suppose f ( X , T 1 , , T s ) Q [ X , T 1 , , X s ] f X , T 1 , … , T s ∈ Q X , T 1 , … , X s f(X,T_(1),dots,T_(s))inQ[X,T_(1),dots,X_(s)]f\left(X, T_{1}, \ldots, T_{s}\right) \in \mathbb{Q}\left[X, T_{1}, \ldots, X_{s}\right]f(X,T1,…,Ts)∈Q[X,T1,…,Xs] is a regular polynomial of total degree d d ddd in the T i T i T_(i)T_{i}Ti and of degree m m mmm in X X XXX with transitive Galois group G S n G ⊂ S n G subS_(n)G \subset S_{n}G⊂Sn over Q ( T 1 , , T s ) Q T 1 , … , T s Q(T_(1),dots,T_(s))\mathbb{Q}\left(T_{1}, \ldots, T_{s}\right)Q(T1,…,Ts). Then | F n ( G ; X ) | f , ε X β ε F n ( G ; X ) ≫ f , ε X β − ε |F_(n)(G;X)|≫_(f,epsi)X^(beta-epsi)\left|\mathscr{F}_{n}(G ; X)\right| \gg_{f, \varepsilon} X^{\beta-\varepsilon}|Fn(G;X)|≫f,εXβ−ε for every ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0, with β = 1 | G | 1 d ( 2 m 2 ) β = 1 − | G | − 1 d ( 2 m − 2 ) beta=(1-|G|^(-1))/(d(2m-2))\beta=\frac{1-|G|^{-1}}{d(2 m-2)}β=1−|G|−1d(2m−2); this is proved in [72]. For G = A n G = A n G=A_(n)G=A_{n}G=An, a polynomial f f fff exhibited by Hilbert can be input to this criterion, implying that | F n ( A n ; X ) | X β n + ε F n A n ; X ≫ X β n + ε |F_(n)(A_(n);X)|≫X^(beta_(n)+epsi)\left|\mathscr{F}_{n}\left(A_{n} ; X\right)\right| \gg X^{\beta_{n}+\varepsilon}|Fn(An;X)|≫Xβn+ε for some β n > 0 β n > 0 beta_(n) > 0\beta_{n}>0βn>0, providing the first lower bound that grows like a power of X X XXX. Here is an open problem: for many groups G G GGG, no such polynomial f f fff has yet been exhibited.
Now we focus on the conjectured upper bound (5.8) for counting fields with bounded discriminant. For any family F = F n ( G ; X ) F = F n ( G ; X ) F=F_(n)(G;X)\mathscr{F}=\mathscr{F}_{n}(G ; X)F=Fn(G;X) of fields, the strong "pointwise" property D F , n ( 0 ) D F , n ( 0 ) D_(F,n)(0)\mathbf{D}_{\mathscr{F}, n}(0)DF,n(0) implies Malle's "average" upper bound (5.8) for the group G G GGG; see [54]. What is more surprising is that there is a converse to this. This relates to our question: can the method of moments be used to deduce the Discriminant Multiplicity Conjecture? Formally, it can. Given a family F F F\mathscr{F}F of fields, for each integer D 1 D ≥ 1 D >= 1D \geq 1D≥1 let m ( D ) m ( D ) m(D)m(D)m(D) denote the number of fields K F K ∈ F K inFK \in \mathscr{F}K∈F with D K = D D K = D D_(K)=DD_{K}=DDK=D. If arbitrarily high k k kkk th moment bounds are known for the function m ( D ) m ( D ) m(D)m(D)m(D), the Discriminant Multiplicity Conjecture follows; see [73]. But the first moment of m ( D ) m ( D ) m(D)m(D)m(D) is the subject of the Malle Conjecture (5.8), so the method of moments certainly seems a difficult avenue to pursue. Yet interestingly, Ellenberg and Venkatesh have shown that in this context the k k kkk th moments can be repackaged as averages.
Informally, the idea is to replace bounding the k k kkk th moment of the function m ( D ) m ( D ) m(D)m(D)m(D) for G G GGG-Galois fields in a family F F F\mathscr{F}F by counting fields in a family F ( k ) F ( k ) F^((k))\mathscr{F}^{(k)}F(k) of G k G k G^(k)G^{k}Gk-Galois fields. Ellenberg and Venkatesh order the fields in F ( k ) F ( k ) F^((k))\mathscr{F}^{(k)}F(k) not by discriminant D K D K D_(K)D_{K}DK, but (roughly speaking) by the square-free kernel D K # D K # D_(K)^(#)D_{K}^{\#}DK# of the discriminant. They generalize the Malle Conjecture to posit that in this ordering, X 1 + ε ≪ X 1 + ε ≪X^(1+epsi)\ll X^{1+\varepsilon}≪X1+ε fields in F ( k ) F ( k ) F(k)\mathscr{F}(k)F(k) have D K # X D K # ≤ X D_(K)^(#) <= XD_{K}^{\#} \leq XDK#≤X, uniformly for all integers k 1 k ≥ 1 k >= 1k \geq 1k≥1. Assuming this conjecture, suppose there are m ( D ) m ( D ) m(D)m(D)m(D) many G G GGG-Galois fields K 1 , , K m ( D ) K 1 , … , K m ( D ) K_(1),dots,K_(m(D))K_{1}, \ldots, K_{m(D)}K1,…,Km(D) with D K i = D D K i = D D_(K_(i))=DD_{K_{i}}=DDKi=D. Taking composita of k k kkk of these generates at least k m ( D ) k ≫ k m ( D ) k ≫_(k)m(D)^(k)\gg_{k} m(D)^{k}≫km(D)k many G k G k G^(k)G^{k}Gk-Galois fields in the family F ( k ) F ( k ) F^((k))\mathscr{F}^{(k)}F(k), with D K # D D K # ≤ D D_(K)^(#) <= DD_{K}^{\#} \leq DDK#≤D. If we suppose m ( D ) D α m ( D ) ≥ D α m(D) >= D^(alpha)m(D) \geq D^{\alpha}m(D)≥Dα for some α > 0 α > 0 alpha > 0\alpha>0α>0 and a sequence of D D → ∞ D rarr ooD \rightarrow \inftyD→∞, under the generalized Malle Conjecture it must be that α k 1 α k ≤ 1 alpha k <= 1\alpha k \leq 1αk≤1 for all k 1 k ≥ 1 k >= 1k \geq 1k≥1. Hence α α alpha\alphaα must be arbitrarily small, as desired.
In full generality, Ellenberg and Venkatesh propose a generalized Malle Conjecture in terms of an f f fff-discriminant, for any rational class function f f fff, and an appropriate generalization a G ( f ) a G ( f ) a_(G)(f)a_{G}(f)aG(f) of the exponent in (5.8). They verify that for a particular choice of f f fff, this implies the Discriminant Multiplicity Conjecture. More recently, Klüners and Wang have shown directly that Malle's Conjecture (5.8) for all groups G G GGG implies the Discriminant Multiplicity Conjecture (also over any number field) [54].
Let us sum up: the upper bound (5.8) in Malle's Conjecture for all groups G G GGG implies the Discriminant Multiplicity Conjecture. The Discriminant Multiplicity Conjecture implies
the â„“ â„“\ellâ„“-torsion Conjecture. Also, the Discriminant Multiplicity Conjecture for F n ( G ; X ) F n ( G ; X ) F_(n)(G;X)\mathscr{F}_{n}(G ; X)Fn(G;X) (that is, property D F , n ( 0 ) ) D F , n ( 0 ) {:D_(F,n)(0))\left.\mathbf{D}_{\mathscr{F}, n}(0)\right)DF,n(0)) implies Malle's Conjecture for F n ( G ; X ) F n ( G ; X ) F_(n)(G;X)\mathscr{F}_{n}(G ; X)Fn(G;X). Moreover, there is one more converse: Alberts has shown that if the â„“ â„“\ellâ„“-torsion Conjecture is true for all solvable extensions and all primes â„“ â„“\ellâ„“ (even just in an average sense), then Malle's upper bound (5.8) holds for all solvable groups [1]. Thus Malle's Conjecture, the Discriminant Multiplicity Conjecture, and the â„“ â„“\ellâ„“-torsion Conjecture are truly equivalent, when restricted to solvable groups. These relationships provide clear motivation for why so many methods described in this survey have involved counting number fields.
In conclusion, we have seen from three different perspectives that the â„“ â„“\ellâ„“-torsion Conjecture should be true. But as Gauss wrote, "Demonstrationes autem rigorosae harum observationum perdifficiles esse videntur."

FUNDING

This work was partially supported by NSF CAREER DMS-1652173.

REFERENCES

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LILLIAN B. PIERCE

Mathematics Department, Duke University, Durham, NC 27708, USA, pierce@ math.duke.edu

POINTS ON SHIMURA VARIETIES MODULO PRIMES

SUG WOO SHIN

ABSTRACT

We survey recent developments on the Langlands-Rapoport conjecture for Shimura varieties modulo primes and an analogous conjecture for Igusa varieties. We discuss resulting implications on the automorphic decomposition of the Hasse-Weil zeta functions and â„“ â„“\ellâ„“ adic cohomology of Shimura varieties, along with further applications to the Langlands correspondence and related problems.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 11G18; Secondary 11F72, 11R39

KEYWORDS

Shimura varieties, Igusa varieties, trace formula, cohomology

1. INTRODUCTION

Shimura varieties have been vital to number theory for their intrinsic beauty and wide-ranging applications. They are simultaneously locally symmetric spaces and quasiprojective varieties over number fields, serving as a geometric bridge between automorphic forms and arithmetic. This feature has been particularly fruitful in the Langlands program.
This paper concentrates on the problem of understanding the Hasse-Weil zeta functions and â„“ â„“\ellâ„“-adic cohomology of Shimura varieties following the approach due to Langlands, Kottwitz, Rapoport, and others. As such, we are naturally led to study integral models and special fibers of Shimura varieties at each prime (Section 2), as epitomized by the LanglandsRapoport (LR) conjecture (Section 3). Below is a partial summary of this article:
Central to this paper is Theorem 3.2, asserting that L R 0 ( S h ) L R 0 ( S h ) LR_(0)(Sh)\mathrm{LR}_{0}(\mathrm{Sh})LR0(Sh), a version of the L R L R LR\mathrm{LR}LR conjecture, is true for Shimura varieties of abelian type with good reduction. This is a strengthening of another version L R 1 ( S h ) L R 1 ( S h ) LR_(1)(Sh)\mathrm{LR}_{1}(\mathrm{Sh})LR1(Sh) which was previously verified by Kisin. Even though L R 0 ( S h ) L R 0 ( S h ) LR_(0)(Sh)\mathrm{LR}_{0}(\mathrm{Sh})LR0(Sh) is weaker than the original LR conjecture (still wide open), it opens doors for most applications. Indeed, the diagram shows how L R 0 ( S h ) L R 0 ( S h ) LR_(0)(Sh)\mathrm{LR}_{0}(\mathrm{Sh})LR0(Sh) implies a (stabilized) trace formula for cohomology of Shimura varieties, designated as T F ( S h ) T F ( S h ) TF(Sh)\mathrm{TF}(\mathrm{Sh})TF(Sh), which in turn leads to interesting applications (Section 4). In Sections 5-6, we survey related problems and directions in the bad reduction case. Finally in Section 7, we review a parallel story for Igusa varieties, where L R 0 ( S h ) L R 0 ( S h ) LR_(0)(Sh)\mathrm{LR}_{0}(\mathrm{Sh})LR0(Sh) provides a key ingredient for proving the analogous assertion L R 0 ( I g ) L R 0 ( I g ) LR_(0)(Ig)\mathrm{LR}_{0}(\mathrm{Ig})LR0(Ig) for Igusa varieties. The dotted vertical arrow suggests that interactions occur between certain applications to Shimura and Igusa varieties, e.g., through Mantovan's formula.

Conventions

Unless otherwise stated, cohomology means the â„“ â„“\ellâ„“-adic étale cohomology with Q ¯ l Q ¯ l bar(Q)_(l)\overline{\mathbb{Q}}_{l}Q¯l coefficients. For an inverse limit of varieties X = ( X i ) X = X i X=(X_(i))X=\left(X_{i}\right)X=(Xi) over a field k k kkk, we write H ( X , Q ¯ l ) H X , Q ¯ l H(X, bar(Q)_(l))H\left(X, \overline{\mathbb{Q}}_{l}\right)H(X,Q¯l) for

cussed results are valid more generally. In the LR conjecture, we omit Z G ( Q p ) Z G Q p Z_(G)(Q_(p))Z_{G}\left(\mathbb{Q}_{p}\right)ZG(Qp)-equivariance to keep the statements simple. If Γ Î“ Gamma\GammaΓ is a topological group, H ( Γ ) H ( Γ ) H(Gamma)\mathscr{H}(\Gamma)H(Γ) is the Hecke algebra of locally constant compactly supported functions on Γ Î“ Gamma\GammaΓ. We write A = Z ^ Z Q A ∞ = Z ^ ⊗ Z Q A^(oo)= hat(Z)ox_(Z)Q\mathbb{A}^{\infty}=\hat{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{Q}A∞=Z^⊗ZQ for the ring of finite adèles, and A , p A ∞ , p A^(oo,p)\mathbb{A}^{\infty, p}A∞,p for the analogous ring without the p p ppp-component. Put Z ˘ p := W ( F ¯ p ) Z ˘ p := W F ¯ p Z^(˘)_(p):=W( bar(F)_(p))\breve{\mathbb{Z}}_{p}:=W\left(\overline{\mathbb{F}}_{p}\right)Z˘p:=W(F¯p) for the Witt ring of F ¯ p F ¯ p bar(F)_(p)\overline{\mathbb{F}}_{p}F¯p, and Q ˘ p := Z ˘ p [ 1 / p ] Q ˘ p := Z ˘ p [ 1 / p ] Q^(˘)_(p):=Z^(˘)_(p)[1//p]\breve{\mathbb{Q}}_{p}:=\breve{\mathbb{Z}}_{p}[1 / p]Q˘p:=Z˘p[1/p]. Denote by σ σ sigma\sigmaσ the Frobenius operator on Q ˘ p Q ˘ p Q^(˘)_(p)\breve{\mathbb{Q}}_{p}Q˘p or a finite unramified extension of Q p Q p Q_(p)\mathbb{Q}_{p}Qp. When we have cohomology spaces H i ( X ) H i ( X ) H^(i)(X)H^{i}(X)Hi(X) (supported on finitely many i i iii 's) with a group action, denote by [ H ( X ) ] = i 0 ( 1 ) i H i ( X ) [ H ( X ) ] = ∑ i ≥ 0   ( − 1 ) i H i ( X ) [H(X)]=sum_(i >= 0)(-1)^(i)H^(i)(X)[H(X)]=\sum_{i \geq 0}(-1)^{i} H^{i}(X)[H(X)]=∑i≥0(−1)iHi(X) the alternating sum viewed in a suitable Grothendieck group of representations. For an algebraic group G G GGG over Q Q Q\mathbb{Q}Q and a field k k kkk over Q Q Q\mathbb{Q}Q, write G k := G × Spec Q G k := G × Spec ⁡ Q G_(k):=Gxx_(Spec Q)G_{k}:=G \times_{\operatorname{Spec} \mathbb{Q}}Gk:=G×Spec⁡Q Spec k k kkk. We quietly fix field
embeddings Q ¯ C , Q ¯ v C Q ¯ ↪ C , Q ¯ v ↪ C bar(Q)↪C, bar(Q)_(v)↪C\overline{\mathbb{Q}} \hookrightarrow \mathbb{C}, \overline{\mathbb{Q}}_{v} \hookrightarrow \mathbb{C}Q¯↪C,Q¯v↪C at each place v v vvv of Q Q Q\mathbb{Q}Q, and identify the residue field of Q ¯ p Q ¯ p bar(Q)_(p)\overline{\mathbb{Q}}_{p}Q¯p with F ¯ p F ¯ p bar(F)_(p)\overline{\mathbb{F}}_{p}F¯p.

2. SHIMURA VARIETIES WITH GOOD REDUCTION

Let G G GGG be a connected reductive group over Q Q Q\mathbb{Q}Q, and X X XXX a G ( R ) G ( R ) G(R)G(\mathbb{R})G(R)-conjugacy class of R R R\mathbb{R}R-group morphisms Res C / R G m G R Res C / R ⁡ G m → G R Res_(C//R)G_(m)rarrG_(R)\operatorname{Res}_{\mathbb{C} / \mathbb{R}} \mathbb{G}_{m} \rightarrow G_{\mathbb{R}}ResC/R⁡Gm→GR. We say that ( G , X ) ( G , X ) (G,X)(G, X)(G,X) is a Shimura datum if it satisfies axioms (2.1.1.1)-(2.1.1.3) of [10]. Each ( G , X ) ( G , X ) (G,X)(G, X)(G,X) determines a conjugacy class of cocharacters μ : G m G C μ : G m → G C mu:G_(m)rarrG_(C)\mu: \mathbb{G}_{m} \rightarrow G_{\mathbb{C}}μ:Gm→GC over C C C\mathbb{C}C, whose field of definition is a number field E = E ( G , X ) C E = E ( G , X ) ⊂ C E=E(G,X)subCE=E(G, X) \subset \mathbb{C}E=E(G,X)⊂C. There is an obvious notion of morphisms between Shimura data.
Thanks to Shimura, Deligne, Borovoi, and Milne, we have a G ( A ) G A ∞ G(A^(oo))G\left(\mathbb{A}^{\infty}\right)G(A∞)-scheme Sh over E E EEE (in the sense of [ 10 , 2.7 .1 ] [ 10 , 2.7 .1 ] [10,2.7.1][10,2.7 .1][10,2.7.1], cf. [31, 1.5.1]), which is a projective limit of quasiprojective varieties over E E EEE with a G ( A ) G A ∞ G(A^(oo))G\left(\mathbb{A}^{\infty}\right)G(A∞)-action. If ( G , X ) ( G , X ) (G,X)(G, X)(G,X) is a Siegel datum, i.e., G = G S p 2 n G = G S p 2 n G=GSp_(2n)G=\mathrm{GSp}_{2 n}G=GSp2n and X X XXX is realized by the Siegel half-spaces of genus n n nnn for some n Z 1 n ∈ Z ≥ 1 n inZ_( >= 1)n \in \mathbb{Z}_{\geq 1}n∈Z≥1, then we obtain (a projective limit of) Siegel modular varieties as output. There is a hierarchy of Shimura data:
(PEL type) (Hodge type) (abelian type) (all).  (PEL type)  ⊂  (Hodge type)  ⊂  (abelian type)  ⊂  (all).  " (PEL type) "sub" (Hodge type) "sub" (abelian type) "sub" (all). "\text { (PEL type) } \subset \text { (Hodge type) } \subset \text { (abelian type) } \subset \text { (all). } (PEL type) ⊂ (Hodge type) ⊂ (abelian type) ⊂ (all). 
Roughly speaking, Shimura varieties coming from PEL-type data are realized as moduli spaces of abelian varieties with polarizations (P), endomorphisms (E), and level (L) structures. 1 1 ^(1){ }^{1}1 This case includes modular curves and, more generally, Siegel modular varieties. A Shimura datum of Hodge type embeds in a Siegel datum by definition, and the corresponding Shimura varieties embed in Siegel modular varieties. Abelian-type data are generalized from those of Hodge type to cover the case when the Dynkin diagram of G Q ¯ G Q ¯ G_( bar(Q))G_{\overline{\mathbb{Q}}}GQ¯ consists of only types A, B, C, and D, with a small exception in the type D case, cf. [10, §2.3].
Now we turn to integral models of Shimura varieties in the good reduction case. A starting point is an unramified Shimura datum ( G , X , p , E ) ( G , X , p , E ) (G,X,p,E)(G, X, p, \mathscr{E})(G,X,p,E), where ( G , X ) ( G , X ) (G,X)(G, X)(G,X) is a Shimura datum, p p ppp is a prime, and E E E\mathscr{E}E is a reductive model of G G GGG over Z p Z p Z_(p)\mathbb{Z}_{p}Zp. The existence of E E E\mathscr{E}E is equivalent to the condition that G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp is an unramified group (i.e., quasisplit over Q p Q p Q_(p)\mathbb{Q}_{p}Qp and split over an unramified extension of Q p ) Q p {:Q_(p))\left.\mathbb{Q}_{p}\right)Qp). Now we put K p := G ( Z p ) K p := G Z p K_(p):=G(Z_(p))K_{p}:=\mathcal{G}\left(\mathbb{Z}_{p}\right)Kp:=G(Zp) and consider the G ( A , p ) G A ∞ , p G(A^(oo,p))G\left(\mathbb{A}^{\infty, p}\right)G(A∞,p)-scheme S h K p S h K p Sh_(K_(p))\mathrm{Sh}_{K_{p}}ShKp over E E EEE, which is similar to Sh as above but has a fixed level K p K p K_(p)K_{p}Kp at p p ppp (while the level subgroup away from p p ppp varies). Kisin [28] ( p > 2 ) ( p > 2 ) (p > 2)(p>2)(p>2) and Kim-Madapusi Pera [27] ( p = 2 ) ( p = 2 ) (p=2)(p=2)(p=2) proved the following fundamental result.
Theorem 2.1. If ( G , X ) ( G , X ) (G,X)(G, X)(G,X) is of abelian type, then there exists a canonical integral model S K p S K p S_(K_(p))\mathscr{S}_{K_{p}}SKp, which is an O E , ( p ) O E , ( p ) O_(E,(p))\mathcal{O}_{E,(p)}OE,(p)-scheme with a G ( A , p ) G A ∞ , p G(A^(oo,p))G\left(\mathbb{A}^{\infty, p}\right)G(A∞,p)-action, such that the generic fiber of S K p S K p S_(K_(p))\mathscr{S}_{K_{p}}SKp is G ( A , p ) G A ∞ , p G(A^(oo,p))G\left(\mathbb{A}^{\infty, p}\right)G(A∞,p)-equivariantly isomorphic to S h K p S h K p Sh_(K_(p))\mathrm{Sh}_{K_{p}}ShKp.
Here "canonical" means that S K p S K p S_(K_(p))\mathscr{S}_{K_{p}}SKp is formally smooth over O E , ( p ) O E , ( p ) O_(E,(p))\mathcal{O}_{E,(p)}OE,(p) and satisfies the extension property of [28, (2.3.7)], which characterizes S K p S K p S_(K_(p))\mathscr{S}_{K_{p}}SKp uniquely up to a unique isomorphism. The proof of the theorem reduces to the Hodge-type case and utilizes the
1 A caveat is that such a moduli space is in general a finite disjoint union of Shimura varieties due to a possible failure of the Hasse principle for G G GGG. See [34, §8] for details.
known canonical integral models in the Siegel case. Kisin constructs S K p S K p S_(K_(p))\mathscr{S}_{K_{p}}SKp by normalizing the closure of S h K p S h K p Sh_(K_(p))\mathrm{Sh}_{K_{p}}ShKp in an ambient Siegel modular variety. The key point is to show formal smoothness of S K p S K p S_(K_(p))\mathscr{S}_{K_{p}}SKp over O E , ( p ) O E , ( p ) O_(E,(p))\mathcal{O}_{E,(p)}OE,(p) by deformation theory and integral p p ppp-adic Hodge theory. The existence of canonical integral models is completely open beyond the abelian-type case.

3. THE LANGLANDS-RAPOPORT CONJECTURE

Given an unramified Shimura datum, the Langlands-Rapoport (LR) conjecture consists of two parts: (i) the existence of canonical integral models and (ii) a group-theoretic description of F ¯ p F ¯ p bar(F)_(p)\overline{\mathbb{F}}_{p}F¯p-points of such integral models. We already addressed (i) in Section 2, which is a prerequisite for discussing (ii) in this section. There is an instructive analogy between (ii) and a description of C C C\mathbb{C}C-points [47, $16]. See Section 6.1 below for the case of bad reduction. We recommend the introduction of [31] for a more detailed survey of the content in this section.

3.1. Galois gerbs

Let k k kkk be a perfect field with an algebraic closure k ¯ k ¯ bar(k)\bar{k}k¯. A Galois gerb over k k kkk consists of a pair ( G , G ) G , G ) G,G)G, G)G,G), where G G GGG is a connected linear algebraic group over k ¯ k ¯ bar(k)\bar{k}k¯, and G G GGG is a topological group extension (with discrete topology on G ( k ¯ ) G ( k ¯ ) G( bar(k))G(\bar{k})G(k¯) and profinite topology on Gal ( k ¯ / k ) Gal ⁡ ( k ¯ / k ) Gal( bar(k)//k)\operatorname{Gal}(\bar{k} / k)Gal⁡(k¯/k) ),
(3.1) 1 G ( k ¯ ) i G π Gal ( k ¯ / k ) 1 , (3.1) 1 → G ( k ¯ ) → i G → Ï€ Gal ⁡ ( k ¯ / k ) → 1 , {:(3.1)1rarr G( bar(k))rarr"i"Grarr"pi"Gal( bar(k)//k)rarr1",":}\begin{equation*} 1 \rightarrow G(\bar{k}) \xrightarrow{i} G \xrightarrow{\pi} \operatorname{Gal}(\bar{k} / k) \rightarrow 1, \tag{3.1} \end{equation*}(3.1)1→G(k¯)→iG→πGal⁡(k¯/k)→1,
such that (i) for every g F g ∈ F g inFg \in \mathbb{F}g∈F, the conjugation by g g ggg on G ( k ¯ ) G ( k ¯ ) G( bar(k))G(\bar{k})G(k¯) is induced by a k ¯ k ¯ bar(k)\bar{k}k¯-group isomorphism π ( g ) G G Ï€ ( g ) ∗ G → ∼ G pi(g)^(**)Grarr"∼"G\pi(g)^{*} G \xrightarrow{\sim} GÏ€(g)∗G→∼G, and (ii) there exists a finite extension K / k K / k K//kK / kK/k in k ¯ k ¯ bar(k)\bar{k}k¯ such that π Ï€ pi\piÏ€ admits a continuous section over Gal ( k ¯ / K ) Gal ⁡ ( k ¯ / K ) Gal( bar(k)//K)\operatorname{Gal}(\bar{k} / K)Gal⁡(k¯/K). If G G GGG is a torus, then (ii) determines a model of G G GGG over k k kkk. We often refer to ( G , G ) ( G , G ) (G,G)(G, G)(G,G) as G G GGG and write Δ â†º Δ ↺Delta\circlearrowleft \Delta↺Δ for G G GGG.
There is a natural notion of morphisms between Galois gerbs over k k kkk. Passing to projective limits, we define pro-Galois gerbs ( G , ( F ) G , ( F ) G,(F)G,(F)G,(F) over k k kkk, which still fit in (3.1) but with G G GGG a pro-algebraic group over k ¯ k ¯ bar(k)\bar{k}k¯. When G G G\mathbb{G}G is a (pro-)Galois gerb over Q Q Q\mathbb{Q}Q, we can localize it at each place v v vvv of Q Q Q\mathbb{Q}Q to obtain a (pro-)Galois gerb over Q v Q v Q_(v)\mathbb{Q}_{v}Qv, to be denoted by G ( v ) G ( v ) G(v)\mathbb{G}(v)G(v).
The most basic example is the neutral Galois gerb F G F G F_(G)\mathscr{F}_{G}FG which arises when G G GGG is already defined over k k kkk. By definition, S G := G ( k ¯ ) Gal ( k ¯ / k ) S G := G ( k ¯ ) ⋊ Gal ⁡ ( k ¯ / k ) S_(G):=G( bar(k))><|Gal( bar(k)//k)\mathbb{S}_{G}:=G(\bar{k}) \rtimes \operatorname{Gal}(\bar{k} / k)SG:=G(k¯)⋊Gal⁡(k¯/k) as a semidirect product with the natural action of Gal ( k ¯ / k ) Gal ⁡ ( k ¯ / k ) Gal( bar(k)//k)\operatorname{Gal}(\bar{k} / k)Gal⁡(k¯/k) on G ( k ) G ( k ) G(k)G(k)G(k).
We introduce a (pro-)Galois gerb G v G v G_(v)\mathbb{G}_{v}Gv over Q v Q v Q_(v)\mathbb{Q}_{v}Qv at each v v vvv. Take জ ∞ জ_(oo)জ_{\infty}জ∞ to be the real Weil group (in particular, F Δ = G m , C F ∞ Δ = G m , C F_(oo)^(Delta)=G_(m,C)\mathcal{F}_{\infty}^{\Delta}=\mathbb{G}_{m, \mathbb{C}}F∞Δ=Gm,C ); the definition of F p F p F_(p)\mathbb{F}_{p}Fp is involved but intended to encode isocrystals. For v p , v ≠ p , ∞ v!=p,oov \neq p, \inftyv≠p,∞, put F v := Gal ( Q ¯ v / Q v ) F v := Gal ⁡ Q ¯ v / Q v F_(v):=Gal( bar(Q)_(v)//Q_(v))\mathbb{F}_{v}:=\operatorname{Gal}\left(\overline{\mathbb{Q}}_{v} / \mathbb{Q}_{v}\right)Fv:=Gal⁡(Q¯v/Qv), namely the trivial neutral Galois gerb.
Central to the LR conjecture is a quasimotivic pro-Galois gerb Q Q Q\mathbb{Q}Q over Q Q Q\mathbb{Q}Q whose algebraic part Q Δ Q Δ Q^(Delta)\mathbb{Q}^{\Delta}QΔ is a pro-torus. The gerb Q Q Q\mathbb{Q}Q comes equipped with morphisms ζ v : G v ζ v : G v → zeta_(v):G_(v)rarr\zeta_{v}: \mathscr{G}_{v} \rightarrowζv:Gv→ { ( v ) ( v ) {(v):}\left\{(v)\right.{(v), and the datum ( { , { ζ v } , ζ v {,{zeta_(v)}:}\left\{,\left\{\zeta_{v}\right\}\right.{,{ζv} ) is uniquely characterized up to a suitable equivalence. A quasimotivic gerb (more precisely, its quotient called a pseudomotivic gerb) is devised as a substitute for the Galois gerb which should arise via Tannaka duality from the category of motives over F ¯ p F ¯ p bar(F)_(p)\overline{\mathbb{F}}_{p}F¯p. The morphisms ζ v ζ v zeta_(v)\zeta_{v}ζv should come from the fiber functors on the latter
category coming from cohomology and polarization structures. See Langlands-Rapoport [41, §§3-4] (complemented by [ 56 , § 8 ] [ 56 , § 8 ] [56,§8][56, \S 8][56,§8] ) and [58, B2.7, B2.8] for further information.
For each torus T T TTT over Q Q Q\mathbb{Q}Q and each cocharacter μ : G m T μ : G m → T mu:G_(m)rarr T\mu: \mathbb{G}_{m} \rightarrow Tμ:Gm→T (defined over a finite extension of Q Q Q\mathbb{Q}Q ), there is a recipe [ 29 [ 29 [29[29[29, (3.1.10)] to define a morphism
(3.2) Ψ T , μ : Q G T (3.2) Ψ T , μ : Q → G T {:(3.2)Psi_(T,mu):QrarrG_(T):}\begin{equation*} \Psi_{T, \mu}: \mathbb{Q} \rightarrow \mathfrak{G}_{T} \tag{3.2} \end{equation*}(3.2)ΨT,μ:Q→GT
As a special case, if ( T , h ) ( T , h ) (T,h)(T, h)(T,h) is a toral Shimura datum, then we obtain Ψ T , μ h Ψ T , μ h Psi_(T,mu_(h))\Psi_{T, \mu_{h}}ΨT,μh with μ h : G m μ h : G m → mu_(h):G_(m)rarr\mu_{h}: \mathbb{G}_{m} \rightarrowμh:Gm→ T T TTT coming from h h hhh. In terms of the heuristics for Q Q Q\mathfrak{Q}Q, the construction of Ψ T , μ h Ψ T , μ h Psi_(T,mu_(h))\Psi_{T, \mu_{h}}ΨT,μh mirrors the operation of taking the mod p p ppp fiber of a C M C M CM\mathrm{CM}CM abelian variety in characteristic 0 .

3.2. Versions of the L R L R LRL RLR conjecture

Let ( G , X , p , E ) ( G , X , p , E ) (G,X,p,E)(G, X, p, \mathscr{E})(G,X,p,E) be an unramified Shimura datum. Write p p p\mathfrak{p}p for the prime of E E EEE over p p ppp, determined by the field embeddings in Section 1, with residue field k ( p ) k ( p ) k(p)k(\mathfrak{p})k(p). A canonical integral model S K p S K p S_(K_(p))\mathscr{S}_{K_{p}}SKp over O E p O E p O_(E_(p))\mathcal{O}_{E_{\mathfrak{p}}}OEp is available in the abelian-type case (Theorem 2.1) and conjectured to exist in general. For the moment, we assume ( G , X ) ( G , X ) (G,X)(G, X)(G,X) to be of Hodge type. Then we can take the partition
(3.3) S K p ( F ¯ p ) = χ I S ( ) (3.3) S K p F ¯ p = ∐ χ ∈ I   S ( â„“ ) {:(3.3)S_(K_(p))( bar(F)_(p))=∐_(chi inI)S(â„“):}\begin{equation*} \mathscr{S}_{K_{p}}\left(\overline{\mathbb{F}}_{p}\right)=\coprod_{\chi \in \mathbb{I}} S(\ell) \tag{3.3} \end{equation*}(3.3)SKp(F¯p)=∐χ∈IS(â„“)
according to the set I I I\mathbb{I}I of isogeny classes, and then parametrize the set S ( l ) S ( l ) S(l)S(\mathscr{l})S(l) consisting of points in each isogeny class â„“ â„“\ellâ„“ relative to a "base point" of choice in â„“ â„“\ellâ„“. This was obtained by Kisin [29, §1.4], where a subtlety in the notion of p p ppp-power isogenies was handled by a result on the connected components of affine Deligne-Lusztig varieties [9]. Each S ( χ ) S ( χ ) S(chi)S(\chi)S(χ) is Φ Z × G ( A , p ) Φ Z × G A ∞ , p Phi^(Z)xx G(A^(oo,p))\Phi^{\mathbb{Z}} \times G\left(\mathbb{A}^{\infty, p}\right)ΦZ×G(A∞,p)-stable, where Φ Î¦ Phi\PhiΦ acts as the geometric Frobenius over k ( p ) k ( p ) k(p)k(\mathfrak{p})k(p), and
(3.4) S ( ) lim K p G ( A , p ) I ( Q ) ( X p ( ) × X p ( d ) / K p ) (3.4) S ( â„“ ) ≅ lim K p ⊂ G A ∞ , p ←   I â„“ ( Q ) ∖ X p ( â„“ ) × X p ( d ) / K p {:(3.4)S(â„“)~=lim_(K^(p)subG(A^(oo),p)^(larr))I_(â„“)(Q)\\(X_(p)(â„“)xxX^(p)(d)//K^(p)):}\begin{equation*} S(\mathcal{\ell}) \cong \lim _{K^{p} \subset \overleftarrow{G\left(\mathbb{A}^{\infty}, p\right)}} I_{\ell}(\mathbb{Q}) \backslash\left(X_{p}(\mathcal{\ell}) \times X^{p}(\mathcal{d}) / K^{p}\right) \tag{3.4} \end{equation*}(3.4)S(â„“)≅limKp⊂G(A∞,p)←Iâ„“(Q)∖(Xp(â„“)×Xp(d)/Kp)
as a right Φ Z × G ( A , p ) Φ Z × G A ∞ , p Phi^(Z)xx G(A^(oo,p))\Phi^{\mathbb{Z}} \times G\left(\mathbb{A}^{\infty, p}\right)ΦZ×G(A∞,p)-set, where X p ( ) X p ( â„“ ) X_(p)(â„“)X_{p}(\mathcal{\ell})Xp(â„“) and X p ( ) X p ( â„“ ) X^(p)(â„“)X^{p}(\mathcal{\ell})Xp(â„“) account for p p ppp-power and prime-to- p p ppp isogenies (from a base point). The quotient by I d ( Q ) I d ( Q ) I_(d)(Q)I_{d}(\mathbb{Q})Id(Q) takes care of redundant counting up to self-isogenies. Since ( G , X ) ( G , X ) (G,X)(G, X)(G,X) is of Hodge type, (3.4) simplifies as I ( Q ) ( X p ( d ) × X p ( d ) ) I â„“ ( Q ) ∖ X p ( d ) × X p ( d ) I_(â„“)(Q)\\(X_(p)(d)xxX^(p)(d))I_{\ell}(\mathbb{Q}) \backslash\left(X_{p}(\mathcal{d}) \times X^{p}(\mathcal{d})\right)Iâ„“(Q)∖(Xp(d)×Xp(d)).
We return to general unramified Shimura data. Following [29, (3.3.6)], one defines admissible morphisms as morphisms ϕ : Q G G Ï• : Q → G G phi:QrarrG_(G)\phi: \mathbb{Q} \rightarrow \mathbb{G}_{G}Ï•:Q→GG satisfying certain conditions to ensure that ϕ Ï• phi\phiÏ• contributes to S K p ( F ¯ p ) S K p F ¯ p S_(K_(p))( bar(F)_(p))\mathscr{S}_{K_{p}}\left(\overline{\mathbb{F}}_{p}\right)SKp(F¯p). In analogy with isogeny classes above, ϕ Ï• phi\phiÏ• gives rise to a right G ( A , p ) G A ∞ , p G(A^(oo,p))G\left(\mathbb{A}^{\infty, p}\right)G(A∞,p)-torsor X p ( ϕ ) X p ( Ï• ) X^(p)(phi)X^{p}(\phi)Xp(Ï•) and a nonempty affine Deligne-Lusztig variety X p ( ϕ ) X p ( Ï• ) X_(p)(phi)X_{p}(\phi)Xp(Ï•) with a Φ Z Φ Z Phi^(Z)\Phi^{\mathbb{Z}}ΦZ-action, where Φ Î¦ Phi\PhiΦ is the Frobenius operator. Write I ϕ I Ï• I_(phi)I_{\phi}IÏ• for the Q Q Q\mathbb{Q}Q-group of automorphisms of ϕ Ï• phi\phiÏ•. Then I ϕ ( A ) I Ï• A ∞ I_(phi)(A^(oo))I_{\phi}\left(\mathbb{A}^{\infty}\right)IÏ•(A∞) naturally acts on X p ( ϕ ) × X p ( ϕ ) X p ( Ï• ) × X p ( Ï• ) X_(p)(phi)xxX^(p)(phi)X_{p}(\phi) \times X^{p}(\phi)Xp(Ï•)×Xp(Ï•). (This is analogous to the self-isogeny of an abelian variety over F ¯ p F ¯ p bar(F)_(p)\overline{\mathbb{F}}_{p}F¯p acting on its étale cohomology away from p p ppp and crystalline cohomology at p p ppp.) Now let I ϕ ad I Ï• ad  I_(phi)^("ad ")I_{\phi}^{\text {ad }}IÏ•ad  denote the Q Q Q\mathbb{Q}Q-group of inner automorphisms of I ϕ I Ï• I_(phi)I_{\phi}IÏ•. Each τ I ϕ ad ( A ) Ï„ ∈ I Ï• ad  A ∞ tau inI_(phi)^("ad ")(A^(oo))\tau \in I_{\phi}^{\text {ad }}\left(\mathbb{A}^{\infty}\right)τ∈IÏ•ad (A∞) can be used to twist the natural action of I ( Q ) I ( Q ) I(Q)I(\mathbb{Q})I(Q) on X p ( ϕ ) × X p ( ϕ ) X p ( Ï• ) × X p ( Ï• ) X_(p)(phi)xxX^(p)(phi)X_{p}(\phi) \times X^{p}(\phi)Xp(Ï•)×Xp(Ï•) :
I ( Q ) I ( A ) τ I ( A ) R X p ( ϕ ) × X p ( ϕ ) . I ( Q ) ⊂ I A ∞ → Ï„ I A ∞ R X p ( Ï• ) × X p ( Ï• ) . I(Q)sub I(A^(oo))rarr"tau"I(A^(oo))RX_(p)(phi)xxX^(p)(phi).I(\mathbb{Q}) \subset I\left(\mathbb{A}^{\infty}\right) \xrightarrow{\tau} I\left(\mathbb{A}^{\infty}\right) \mathbb{R} X_{p}(\phi) \times X^{p}(\phi) .I(Q)⊂I(A∞)→τI(A∞)RXp(Ï•)×Xp(Ï•).
Taking the left quotient by this action (denoted τ ∖ Ï„ \\_(tau)\backslash_{\tau}∖τ below), we define a Φ Z × G ( A , p ) Φ Z × G A ∞ , p Phi^(Z)xx G(A^(oo,p))\Phi^{\mathbb{Z}} \times G\left(\mathbb{A}^{\infty, p}\right)ΦZ×G(A∞,p)-set
(3.5) S τ ( ϕ ) := lim K p G ( A , p ) I ϕ ( Q ) τ ( X p ( ϕ ) × X p ( ϕ ) / K p ) (3.5) S Ï„ ( Ï• ) := lim K p ⊂ G A ∞ , p ←   I Ï• ( Q ) ∖ Ï„ X p ( Ï• ) × X p ( Ï• ) / K p {:(3.5)S_(tau)(phi):=lim_(K^(p)subG(A^(oo),p)^(larr))I_(phi)(Q)\\_(tau)(X_(p)(phi)xxX^(p)(phi)//K^(p)):}\begin{equation*} S_{\tau}(\phi):=\lim _{K^{p} \subset \overleftarrow{G\left(\mathbb{A}^{\infty}, p\right)}} I_{\phi}(\mathbb{Q}) \backslash_{\tau}\left(X_{p}(\phi) \times X^{p}(\phi) / K^{p}\right) \tag{3.5} \end{equation*}(3.5)SÏ„(Ï•):=limKp⊂G(A∞,p)←IÏ•(Q)∖τ(Xp(Ï•)×Xp(Ï•)/Kp)
We just write S ( ϕ ) S ( Ï• ) S(phi)S(\phi)S(Ï•) if τ Ï„ tau\tauÏ„ is trivial. The isomorphism class of S τ ( ϕ ) S Ï„ ( Ï• ) S_(tau)(phi)S_{\tau}(\phi)SÏ„(Ï•) depends only on [ τ ] [ Ï„ ] ∈ [tau]in[\tau] \in[Ï„]∈ H ( ϕ ) := I ϕ ad ( Q ) I ϕ ad ( A ) / I ϕ ( A ) H ( Ï• ) := I Ï• ad  ( Q ) ∖ I Ï• ad  A ∞ / I Ï• A ∞ H(phi):=I_(phi)^("ad ")(Q)\\I_(phi)^("ad ")(A^(oo))//I_(phi)(A^(oo))\mathscr{H}(\phi):=I_{\phi}^{\text {ad }}(\mathbb{Q}) \backslash I_{\phi}^{\text {ad }}\left(\mathbb{A}^{\infty}\right) / I_{\phi}\left(\mathbb{A}^{\infty}\right)H(Ï•):=IÏ•ad (Q)∖IÏ•ad (A∞)/IÏ•(A∞) represented by τ Ï„ tau\tauÏ„. (The right quotient is taken with respect to the multiplication through the natural map I ϕ I ϕ ad I Ï• → I Ï• ad  I_(phi)rarrI_(phi)^("ad ")I_{\phi} \rightarrow I_{\phi}^{\text {ad }}Iϕ→IÏ•ad .) If ϕ , ϕ Ï• , Ï• ′ phi,phi^(')\phi, \phi^{\prime}Ï•,ϕ′ are G ( Q ¯ ) G ( Q ¯ ) G( bar(Q))G(\overline{\mathbb{Q}})G(Q¯)-conjugate, then S τ ( ϕ ) S τ ( ϕ ) S Ï„ ( Ï• ) ≅ S Ï„ Ï• ′ S_(tau)(phi)~=S_(tau)(phi^('))S_{\tau}(\phi) \cong S_{\tau}\left(\phi^{\prime}\right)SÏ„(Ï•)≅SÏ„(ϕ′) and canonically H ( ϕ ) H ( ϕ ) H ( Ï• ) ≅ H Ï• ′ H(phi)~=H(phi^('))\mathscr{H}(\phi) \cong \mathscr{H}\left(\phi^{\prime}\right)H(Ï•)≅H(ϕ′). Denoting by J J J\mathbb{J}J the set of G ( Q ¯ ) G ( Q ¯ ) G( bar(Q))G(\overline{\mathbb{Q}})G(Q¯) conjugacy classes of admissible morphisms, we write H ( L ) H ( L ) H(L)\mathscr{H}(\mathcal{L})H(L) and S τ ( L ) S Ï„ ( L ) S_(tau)(L)S_{\tau}(\mathcal{L})SÏ„(L) respectively for H ( ϕ ) H ( Ï• ) H(phi)\mathscr{H}(\phi)H(Ï•) and S τ ( ϕ ) S Ï„ ( Ï• ) S_(tau)(phi)S_{\tau}(\phi)SÏ„(Ï•), when g g g\mathscr{g}g is the G ( Q ¯ ) G ( Q ¯ ) G( bar(Q))G(\overline{\mathbb{Q}})G(Q¯)-conjugacy class of ϕ Ï• phi\phiÏ•.
It is convenient to name a "rationality" condition on the adelic element τ I ϕ ad ( A ) Ï„ ∈ I Ï• ad  A ∞ tau inI_(phi)^("ad ")(A^(oo))\tau \in I_{\phi}^{\text {ad }}\left(\mathbb{A}^{\infty}\right)τ∈IÏ•ad (A∞) that is technical but useful. For each maximal torus T T TTT of I ϕ I Ï• I_(phi)I_{\phi}IÏ• over Q Q Q\mathbb{Q}Q, we have the maps
I ϕ a d ( A ) H 1 ( A , Z I ϕ ) H 1 ( A , T ) I Ï• a d A ∞ → ∂ H 1 A ∞ , Z I Ï• → H 1 A ∞ , T I_(phi)^(ad)(A^(oo))rarr"del"H^(1)(A^(oo),Z_(I_(phi)))rarrH^(1)(A^(oo),T)I_{\phi}^{\mathrm{ad}}\left(\mathbb{A}^{\infty}\right) \xrightarrow{\partial} H^{1}\left(\mathbb{A}^{\infty}, Z_{I_{\phi}}\right) \rightarrow H^{1}\left(\mathbb{A}^{\infty}, T\right)IÏ•ad(A∞)→∂H1(A∞,ZIÏ•)→H1(A∞,T)
where ∂ del\partial∂ is the connecting homomorphism, and the second map is induced by Z I ϕ T Z I Ï• ⊂ T Z_(I_(phi))sub TZ_{I_{\phi}} \subset TZIϕ⊂T. We say that τ Ï„ tau\tauÏ„ is tori-rational if the image of τ Ï„ tau\tauÏ„ in H 1 ( A , T ) H 1 A ∞ , T H^(1)(A^(oo),T)H^{1}\left(\mathbb{A}^{\infty}, T\right)H1(A∞,T) lies in the subset of the image of H 1 ( Q , T ) H 1 ( A , T ) H 1 ( Q , T ) → H 1 A ∞ , T H^(1)(Q,T)rarrH^(1)(A^(oo),T)H^{1}(\mathbb{Q}, T) \rightarrow H^{1}\left(\mathbb{A}^{\infty}, T\right)H1(Q,T)→H1(A∞,T) which maps trivially into the abelianized cohomology of G G GGG, for every T T TTT. This condition depends only on [ τ ] H ( ϕ ) [ Ï„ ] ∈ H ( Ï• ) [tau]inH(phi)[\tau] \in \mathscr{H}(\phi)[Ï„]∈H(Ï•).
We are ready to state versions of the LR conjecture in increasing order of strength. (To be precise, the conjecture requires extra compatibility conditions on τ ( J ) Ï„ ( J ) tau(J)\tau(\mathcal{J})Ï„(J) under cohomological twistings of g g g\mathscr{g}g in ( L R 1 ) L R 1 (LR_(1))\left(\mathrm{LR}_{1}\right)(LR1) and ( L R 0 ) L R 0 (LR_(0))\left(\mathrm{LR}_{0}\right)(LR0), but we avoid mentioning them explicitly in this exposition. See [31, §$2.6-2.7], where these conditions correspond to τ _ Γ ( H ) 1 Ï„ _ ∈ Γ ( H ) 1 tau _in Gamma(H)_(1)\underline{\tau} \in \Gamma(\mathscr{H})_{1}Ï„_∈Γ(H)1 and τ _ Γ ( H ) 0 Ï„ _ ∈ Γ ( H ) 0 tau _in Gamma(H)_(0)\underline{\tau} \in \Gamma(\mathscr{H})_{0}Ï„_∈Γ(H)0, respectively. With this correction, the Langlands-Rapoport- τ Ï„ tau\tauÏ„ conjecture therein is exactly ( L R 0 ) L R 0 (LR_(0))\left(\mathrm{LR}_{0}\right)(LR0) below.)
Conjecture 3.1. The following assertions hold true:
( L R 1 ) L R 1 (LR_(1))\left(\mathrm{LR}_{1}\right)(LR1) There exists a Φ Z × G ( A , p ) Φ Z × G A ∞ , p Phi^(Z)xx G(A^(oo,p))\Phi^{\mathbb{Z}} \times G\left(\mathbb{A}^{\infty, p}\right)ΦZ×G(A∞,p)-equivariant bijection
S K p ( F ¯ p ) L J S τ ( J ) ( J ) S K p F ¯ p ≅ ∐ L ∈ J   S Ï„ ( J ) ( J ) S_(K_(p))( bar(F)_(p))~=∐_(LinJ)S_(tau(J))(J)\mathscr{S}_{K_{p}}\left(\overline{\mathbb{F}}_{p}\right) \cong \coprod_{\mathcal{L} \in \mathbb{J}} S_{\tau(\mathcal{J})}(\mathcal{J})SKp(F¯p)≅∐L∈JSÏ„(J)(J)
for some family of elements { τ ( J ) H ( d ) } L J { Ï„ ( J ) ∈ H ( d ) } L ∈ J {tau(J)inH(d)}_(LinJ)\{\tau(\mathcal{J}) \in \mathscr{H}(\mathcal{d})\}_{\mathcal{L} \in \mathbb{J}}{Ï„(J)∈H(d)}L∈J.
( L R 0 ) L R 0 (LR_(0))\left(\mathrm{LR}_{0}\right)(LR0) The conclusion of ( L R 1 ) L R 1 (LR_(1))\left(\mathrm{LR}_{1}\right)(LR1) holds with τ ( H ) Ï„ ( H ) tau(H)\tau(\mathcal{H})Ï„(H) tori-rational for every G J G ∈ J GinJ\mathcal{G} \in \mathbb{J}G∈J.
(LR) The conclusion of ( L R 1 ) L R 1 (LR_(1))\left(\mathrm{LR}_{1}\right)(LR1) holds with τ ( H ) Ï„ ( H ) tau(H)\tau(\mathcal{H})Ï„(H) trivial for every J J J ∈ J JinJ\mathcal{J} \in \mathbb{J}J∈J.
Statement (LR) is nothing but the original LR conjecture. In the Hodge-type case (to which the abelian-type case can be reduced in practice), a natural approach in view of (3.3) is to establish a bijection J I L J J ∈ I ↔ L ∈ J JinIharrLinJ\mathcal{J} \in \mathbb{I} \leftrightarrow \mathcal{L} \in \mathbb{J}J∈I↔L∈J such that there exists a Φ Z × G ( A , p ) Φ Z × G A ∞ , p Phi^(Z)xx G(A^(oo,p))\Phi^{\mathbb{Z}} \times G\left(\mathbb{A}^{\infty, p}\right)ΦZ×G(A∞,p)-equivariant bijection S ( ) S τ ( J ) ( J ) S ( â„“ ) ≅ S Ï„ ( J ) ( J ) S(â„“)~=S_(tau(J))(J)S(\mathcal{\ell}) \cong S_{\tau(\mathcal{J})}(\mathcal{J})S(â„“)≅SÏ„(J)(J) with constraints on τ ( J ) Ï„ ( J ) tau(J)\tau(\mathcal{J})Ï„(J) as in the conjecture.
It is known ([31, §3], cf. [42, 46]) that (LR) implies (3.7) below, which is the gateway to applications, but (LR) remains to be completely open even for Siegel modular varieties (of genus 2 ≥ 2 >= 2\geq 2≥2 ). Milne proved that the original LR conjecture follows from the Hodge conjecture for C M C M CM\mathrm{CM}CM abelian varieties (see [48, P. 4] and the references therein), but the latter conjecture is also wide open.
On a positive note, Kisin [29] made a major breakthrough to prove ( L R 1 ) L R 1 (LR_(1))\left(\mathrm{LR}_{1}\right)(LR1) for all unramified Shimura data ( G , X , p , E ) ( G , X , p , E ) (G,X,p,E)(G, X, p, \mathcal{E})(G,X,p,E) of abelian type. Unfortunately, ( L R 1 ) L R 1 (LR_(1))\left(\mathrm{LR}_{1}\right)(LR1) by itself is not strong enough for the next steps. This motivated us to formulate and prove the strengthening
( L R 0 ) L R 0 (LR_(0))\left(\mathrm{LR}_{0}\right)(LR0) in [31], which suffices for the trace formula (Section 3.3) and applications (Section 4) below.
Theorem 3.2. For every unramified Shimura datum ( G , X , p , E ) ( G , X , p , E ) (G,X,p,E)(G, X, p, \mathcal{E})(G,X,p,E) of abelian type, Conjecture ( L R 0 ) L R 0 (LR_(0))\left(\mathrm{LR}_{0}\right)(LR0) holds true.
Let us sketch some ideas of proof when ( G , X ) ( G , X ) (G,X)(G, X)(G,X) is of Hodge type. The reduction to this case is nontrivial and convoluted, cf. [31, $6]. Already in [29], Kisin proved a refinement of ( L R 1 ) L R 1 (LR_(1))\left(\mathrm{LR}_{1}\right)(LR1) in order to propagate ( L R 1 ) L R 1 (LR_(1))\left(\mathrm{LR}_{1}\right)(LR1) through Deligne's formalism of connected Shimura varieties. With that said, we focus on the Hodge-type setting for simplicity.
The proof consists of two parts: (i) constructing a bijection d I d J d ∈ I ↔ d ∈ J dinIharrdinJ\mathscr{d} \in \mathbb{I} \leftrightarrow \mathscr{d} \in \mathbb{J}d∈I↔d∈J and (ii) showing that S ( d ) S τ ( J ) ( d ) S ( d ) ≅ S Ï„ ( J ) ( d ) S(d)~=S_(tau(J))(d)S(\mathcal{d}) \cong S_{\tau(\mathcal{J})}(\mathcal{d})S(d)≅SÏ„(J)(d) with some control over τ ( d ) Ï„ ( d ) tau(d)\tau(\mathcal{d})Ï„(d). A crucial idea is to use special point data, namely toral Shimura data ( T , h T T , h T (T,h_(T):}\left(T, h_{T}\right.(T,hT ) with embeddings into ( G , X ) ( G , X ) (G,X)(G, X)(G,X), to probe both sides of the bijection. Such data can be mapped into I I I\mathbb{I}I by taking mod p p ppp of the corresponding special points on S h K p S h K p Sh_(K_(p))\mathrm{Sh}_{K_{p}}ShKp, and to J J J\mathbb{J}J by composing (3.2) with the induced embedding F T F G F T ↪ F G F_(T)↪F_(G)\mathfrak{F}_{T} \hookrightarrow \mathfrak{F}_{G}FT↪FG. The map to I I I\mathbb{I}I is onto by Kisin [29], generalizing Honda's result on C M C M CM\mathrm{CM}CM lifting of an abelian variety over F ¯ p F ¯ p bar(F)_(p)\overline{\mathbb{F}}_{p}F¯p up to isogeny. The surjectivity onto J J J\mathbb{J}J is due to LanglandsRapoport [41]:
From each of I I I\mathbb{I}I and J J J\mathbb{J}J, Kisin [29] constructed Kottwitz triples consisting of certain conjugacy classes on G G GGG up to an equivalence, and showed that the outer diagram above commutes. This determines a bijection I J I ≅ J I~=J\mathbb{I} \cong \mathbb{J}I≅J up to a finite ambiguity since the maps to Kottwitz triples have finite fibers. However, S ( d ) S ( d ) S(d)S(\mathscr{d})S(d) and S ( F ) S ( F ) S(F)S(\mathcal{F})S(F) need not be isomorphic through this bijection, since Kottwitz triples forget part of their structures. The possible deviation is recorded by τ ( J ) Ï„ ( J ) tau(J)\tau(\mathscr{J})Ï„(J), which is a priori under little control. This is still enough for deducing ( L R 1 ) L R 1 (LR_(1))\left(\mathrm{LR}_{1}\right)(LR1).
To prove ( L R 0 ) L R 0 (LR_(0))\left(\mathrm{LR}_{0}\right)(LR0), various refinements and improvements are made on both (i) and (ii) of the argument. Since Shimura varieties from toral Shimura data have canonical base points, a special point datum not only determines d I d ∈ I d inId \in \mathbb{I}d∈I but also a distinguished point on S ( d ) S ( d ) S(d)S(\mathcal{d})S(d). Similarly, we have a base point on S ( J ) S ( J ) S(J)S(\mathcal{J})S(J) as well if J J J\mathcal{J}J comes from the same special point datum. The two points on S ( ) S ( ℓ ) S(ℓ)S(\mathscr{\ell})S(ℓ) and S ( J ) S ( J ) S(J)S(\mathcal{J})S(J) are difficult to relate, but they are shown to be compatible on the level of the maximal abelian quotient G a b G a b G^(ab)G^{\mathrm{ab}}Gab, based on integral p p ppp-adic Hodge theory of crystalline lattices in G G GGG-valued Galois representations, among other things. (A relevant technical issue is that the Q p Q p Q_(p)\mathbb{Q}_{p}Qp-embedding T G T ↪ G T↪GT \hookrightarrow GT↪G does not extend to a Z p Z p Z_(p)\mathbb{Z}_{p}Zp-map from the Néron model of T T TTT to E E E\mathscr{E}E over Z p Z p Z_(p)\mathbb{Z}_{p}Zp, but this is fine if G , E G , E G,EG, \mathcal{E}G,E are replaced with G a b , E a b G a b , E a b G^(ab),E^(ab)G^{\mathrm{ab}}, \mathcal{E}^{\mathrm{ab}}Gab,Eab.) Further arguments (sketched in [ 31 , $ 0.5 ] ) [ 31 , $ 0.5 ] ) [31,$0.5])[31, \$ 0.5])[31,$0.5]) amplify this compatibility to ( L R 0 ) L R 0 (LR_(0))\left(\mathrm{LR}_{0}\right)(LR0).

3.3. From the LR conjecture to a stabilized trace formula

Here we return to a general unramified Shimura datum ( G , X , p , E ) ( G , X , p , E ) (G,X,p,E)(G, X, p, \mathcal{E})(G,X,p,E) (possibly not of abelian type). Set r Z r ∈ Z r inZr \in \mathbb{Z}r∈Z to be the inertia degree of p p p\mathfrak{p}p over p p ppp. As indicated above, ( L R 0 ) L R 0 {:LR_(0))\left.\mathrm{LR}_{0}\right)LR0)
is designed as a substitute for (LR) to imply the following formula predicted by [ 32 , 41 ] [ 32 , 41 ] [32,41][32,41][32,41]. The implication is shown in [31, §3] (refer to the latter for undefined notation):
tr ( f , p × Φ p j [ H c ( Sh K p , Q ¯ l ) ] ) = c c ( c ) O γ ( c ) ( f , p ) T O δ ( c ) ( ϕ ( j ) ) , j 1 tr ⁡ f ∞ , p × Φ p j ∣ H c Sh K p , Q ¯ l = ∑ c   c ( c ) O γ ( c ) f ∞ , p T O δ ( c ) Ï• ( j ) , j ≫ 1 tr(f^(oo,p)xxPhi_(p)^(j)∣[H_(c)(Sh_(K_(p)), bar(Q)_(l))])=sum_(c)c(c)O_(gamma(c))(f^(oo,p))TO_(delta(c))(phi^((j))),quad j≫1\operatorname{tr}\left(f^{\infty, p} \times \Phi_{\mathfrak{p}}^{j} \mid\left[H_{c}\left(\operatorname{Sh}_{K_{p}}, \overline{\mathbb{Q}}_{l}\right)\right]\right)=\sum_{\mathfrak{c}} c(\mathfrak{c}) O_{\gamma(\mathfrak{c})}\left(f^{\infty, p}\right) T O_{\delta(\mathfrak{c})}\left(\phi^{(j)}\right), \quad j \gg 1tr⁡(f∞,p×Φpj∣[Hc(ShKp,Q¯l)])=∑cc(c)Oγ(c)(f∞,p)TOδ(c)(Ï•(j)),j≫1
Here ϕ ( j ) Ï• ( j ) phi^((j))\phi^{(j)}Ï•(j) is an explicit function in the unramified Hecke algebra of G ( Q p j r ) G Q p j r G(Q_(p)^(jr))G\left(\mathbb{Q}_{p}{ }^{j r}\right)G(Qpjr) (with respect to G ( Z p j r ) G Z p j r G(Z_(p^(jr)))\mathcal{G}\left(\mathbb{Z}_{p^{j r}}\right)G(Zpjr) ), and the sum runs over certain group-theoretic data c (called Kottwitz parameters) fibered over the set of stable conjugacy classes in G ( Q ) G ( Q ) G(Q)G(\mathbb{Q})G(Q) which are elliptic in G ( R ) G ( R ) G(R)G(\mathbb{R})G(R). Here c determines an explicit constant c ( c ) Q , γ ( c ) G ( A , p ) c ( c ) ∈ Q , γ ( c ) ∈ G A ∞ , p c(c)inQ,gamma(c)in G(A^(oo,p))c(\mathfrak{c}) \in \mathbb{Q}, \gamma(\mathfrak{c}) \in G\left(\mathbb{A}^{\infty, p}\right)c(c)∈Q,γ(c)∈G(A∞,p) up to conjugacy, and δ ( c ) δ ( c ) ∈ delta(c)in\delta(\mathrm{c}) \inδ(c)∈ G ( Q p j r ) G Q p j r G(Q_(p^(jr)))G\left(\mathbb{Q}_{p^{j r}}\right)G(Qpjr) up to σ σ sigma\sigmaσ-conjugacy. In particular, the orbital integral O γ ( c ) ( f , p ) O γ ( c ) f ∞ , p O_(gamma(c))(f^(oo,p))O_{\gamma(\mathfrak{c})}\left(f^{\infty, p}\right)Oγ(c)(f∞,p) on G ( A , p ) G A ∞ , p G(A^(oo,p))G\left(\mathbb{A}^{\infty, p}\right)G(A∞,p) and the σ σ sigma\sigmaσ-twisted orbital integral T O δ ( c ) ( ϕ ( j ) ) T O δ ( c ) Ï• ( j ) TO_(delta(c))(phi^((j)))T O_{\delta(c)}\left(\phi^{(j)}\right)TOδ(c)(Ï•(j)) on G ( Q p j r ) G Q p j r G(Q_(p^(jr)))G\left(\mathbb{Q}_{p^{j r}}\right)G(Qpjr) are well defined. Stabilizing the right-hand side, we arrive at the following, which is a rough version of [31, THM. 3 AND 4].
Theorem 3.3. Assume that ( L R 0 ) L R 0 (LR_(0))\left(\mathrm{LR}_{0}\right)(LR0) is true. For every f , p H ( G ( A , p ) ) f ∞ , p ∈ H G A ∞ , p f^(oo,p)inH(G(A^(oo,p)))f^{\infty, p} \in \mathscr{H}\left(G\left(\mathbb{A}{ }^{\infty, p}\right)\right)f∞,p∈H(G(A∞,p)), there exists a constant j 0 j 0 j_(0)j_{0}j0 such that for every j j 0 j ≥ j 0 j >= j_(0)j \geq j_{0}j≥j0, a formula of the following form holds.
(3.8) tr ( f , p × Φ j [ H c ( Sh K p , Q ¯ l ) ] ) = e E e l l ( G ) ST e l l e ( f e , , p f p e , ( j ) f e ) (3.8) tr ⁡ f ∞ , p × Φ j ∣ H c Sh K p , Q ¯ l = ∑ e ∈ E e l l ( G )   ST e l l e ⁡ f e , ∞ , p f p e , ( j ) f ∞ e {:(3.8)tr(f^(oo,p)xxPhi^(j)∣[H_(c)(Sh_(K_(p)), bar(Q)_(l))])=sum_(e inE_(ell)(G))ST_(ell)^(e)(f^(e,oo,p)f_(p)^(e,(j))f_(oo)^(e)):}\begin{equation*} \operatorname{tr}\left(f^{\infty, p} \times \Phi^{j} \mid\left[H_{c}\left(\operatorname{Sh}_{K_{p}}, \overline{\mathbb{Q}}_{l}\right)\right]\right)=\sum_{e \in \mathcal{E}_{\mathrm{ell}}(G)} \operatorname{ST}_{\mathrm{ell}}^{e}\left(f^{e, \infty, p} f_{p}^{e,(j)} f_{\infty}^{e}\right) \tag{3.8} \end{equation*}(3.8)tr⁡(f∞,p×Φj∣[Hc(ShKp,Q¯l)])=∑e∈Eell(G)STelle⁡(fe,∞,pfpe,(j)f∞e)
where E e l l ( G ) E e l l ( G ) E_(ell)(G)\mathcal{E}_{\mathrm{ell}}(G)Eell(G) is the set of elliptic endoscopic data for G G GGG up to isomorphism, and S T e l l e S T e l l e ST_(ell)^(e)\mathrm{ST}_{\mathrm{ell}}^{e}STelle is the stable elliptic distribution associated with the endoscopic datum e e eee.
In light of Theorem 3.2, the conclusion of the theorem is unconditionally true for ( G , X ) ( G , X ) (G,X)(G, X)(G,X) of abelian type. We can easily allow a nonconstant coefficient as done in [31].
The proof of (3.7) from ( L R 0 ) L R 0 (LR_(0))\left(\mathrm{LR}_{0}\right)(LR0) is mostly close to the deduction from (LR) (cf. [46]), and starts from the fixed-point formula for (improper) varieties over finite fields due to Fujiwara and Varshavsky [13,70]; this explains the condition on j j jjj. To compute the cohomology of the generic fiber via that of the special fiber, we apply Lan-Stroh's result [39]. Tori-rationality in ( L R 0 ) L R 0 (LR_(0))\left(\mathrm{LR}_{0}\right)(LR0) is the main point to ensure that the fixed-point counting is not affected by the presence of τ ( F ) Ï„ ( F ) tau(F)\tau(\mathcal{F})Ï„(F) even if τ ( L ) Ï„ ( L ) tau(L)\tau(\mathcal{L})Ï„(L) is nontrivial. The stabilization from (3.7) to Theorem 3.3 follows the argument in [32] with small improvements to work without technical hypotheses. We note that f e , , p f e , ∞ , p f^(e,oo,p)f^{e, \infty, p}fe,∞,p is the Langlands-Shelstad transfer of f , p f ∞ , p f^(oo,p)f^{\infty, p}f∞,p whereas f p e , ( j ) f p e , ( j ) f_(p)^(e,(j))f_{p}^{e,(j)}fpe,(j) and f e f ∞ e f_(oo)^(e)f_{\infty}^{e}f∞e are constructed differently. (See [31, §8.2].) As usual in endoscopy, auxiliary z z zzz-extensions are chosen if the derived subgroup of G G GGG is not simply connected, and the right-hand side of (3.8) should be interpreted appropriately.
Remark 3.4. Sometimes it is possible to obtain (3.8) bypassing any version of the LR conjecture. When ( G , X ) ( G , X ) (G,X)(G, X)(G,X) is of PEL type A or C, this is done by Kottwitz [34]; for Hodge-type data, this is worked out by Lee [42]. It is unclear how their methods interact with connected components of Shimura varieties, so their results do not easily extend to the abelian-type setup. In contrast, the formalism of the LR conjecture is well suited to such extensions.
Remark 3.5. If the adjoint quotient G / Z G G / Z G G//Z_(G)G / Z_{G}G/ZG is isotropic over Q Q Q\mathbb{Q}Q or, equivalently, if Sh is not proper over E E EEE (at each fixed level), it is desirable to prove the analogue of (3.8) for the intersection cohomology of the Satake-Baily-Borel compactification; see [51, §84-5] for what
new problems need to be solved. This has been carried out for certain unitary and orthogonal Shimura varieties, as well as Siegel modular varieties, in [ 50 , 52 , 73 ] [ 50 , 52 , 73 ] [50,52,73][50,52,73][50,52,73].

4. APPLICATIONS

4.1. The Hasse-Weil zeta functions and â„“ â„“\ellâ„“-adic cohomology

As pioneered by Eichler, Shimura, Deligne, Kuga, Sato, and Ihara, a central problem on Shimura varieties is to compute their ζ ζ zeta\zetaζ-functions and â„“ â„“\ellâ„“-adic cohomology. The goals are (i) to express the ζ ζ zeta\zetaζ-function as a quotient of products of automorphic L L LLL-functions (thereby deduce a meromorphic continuation and a functional equation when the L L LLL-functions are sufficiently understood), cf. [6, coNJ. 5.2], and (ii) to decompose the â„“ â„“\ellâ„“-adic cohomology according to automorphic representations and identify the Galois action on each piece. To this end, Langlands and Kottwitz developed a robust method in a series of papers in the 1970-1980s (from [40] to [32]). At the heart is a comparison between the Arthur-Selberg trace formula and a conjectural trace formula for the Hecke-Frobenius action on the cohomology at good primes p p ppp, where the latter should come from a fixed-point formula for the special fiber of Shimura varieties modulo p p ppp.
When G / Z G G / Z G G//Z_(G)G / Z_{G}G/ZG is anisotropic over Q Q Q\mathbb{Q}Q (equivalently, when Sh is an inverse limit of projective varieties), Theorem 3.3 should be sufficient for the goals (i) and (ii) (up to semisimplifying the Galois action), by following the outline in [32, 888 10 ] 888 − 10 ] 888-10]888-10]888−10]. We say "should" for two reasons. Firstly, we do not have enough knowledge about automorphic representations in general (e.g., endoscopic classification, cf. [32, §8]). Thus complete details have not been worked out apart from low-rank examples, some special cases such as [33], or under simplifying hypotheses. Secondly, we typically need a positive answer to the following problem to proceed. 2 2 ^(2){ }^{2}2 The reason is that S T e S T e ST^(e)\mathrm{ST}^{e}STe should admit a relatively clean spectral expansion in terms of the discrete automorphic spectrum of endoscopic groups for G G GGG, but the spectral interpretation of S T ell e S T ell  e ST_("ell ")^(e)\mathrm{ST}_{\text {ell }}^{e}STell e is expected to be quite complicated in general.
Problem 4.1. Assume that G / Z G G / Z G G//Z_(G)G / Z_{G}G/ZG is Q Q Q\mathbb{Q}Q-anisotropic. In (3.8), prove that
ST e l l e ( f e , , p f p e , ( j ) f e ) = ST e ( f e , , p f p e , ( j ) f e ) , e E e l l ( G ) ST e l l e ⁡ f e , ∞ , p f p e , ( j ) f ∞ e = ST e ⁡ f e , ∞ , p f p e , ( j ) f ∞ e , ∀ e ∈ E e l l ( G ) ST_(ell)^(e)(f^(e,oo,p)f_(p)^(e,(j))f_(oo)^(e))=ST^(e)(f^(e,oo,p)f_(p)^(e,(j))f_(oo)^(e)),quad AA e inE_(ell)(G)\operatorname{ST}_{\mathrm{ell}}^{e}\left(f^{e, \infty, p} f_{p}^{e,(j)} f_{\infty}^{e}\right)=\operatorname{ST}^{e}\left(f^{e, \infty, p} f_{p}^{e,(j)} f_{\infty}^{e}\right), \quad \forall e \in \mathcal{E}_{\mathrm{ell}}(G)STelle⁡(fe,∞,pfpe,(j)f∞e)=STe⁡(fe,∞,pfpe,(j)f∞e),∀e∈Eell(G)
where S T e S T e ST^(e)\mathrm{ST}^{e}STe stands for the stable distribution as defined in [ 52 , $ 5.4 ] [ 52 , $ 5.4 ] [52,$5.4][52, \$ 5.4][52,$5.4].
Although this problem is open, there are quite a few examples where it is known either by the nature of G G GGG or under a simplifying hypothesis on the test function. This provides a starting point for the Langlands correspondence (Section 4.2 below).
Now we remove the assumption on G / Z G G / Z G G//Z_(G)G / Z_{G}G/ZG. In fact, the argument outlined in [32, $ 8 8 1 0 ] $ 8 8 − 1 0 ] $88-10]\$ 8 \mathbf{8 - 1 0 ]}$88−10] is given in this generality, conditional on an affirmative answer to the following.
2 A shortcut getting around Problem 4.1 is possible when G G GGG has "no endoscopy," e.g., if G G GGG is a form of G L 2 G L 2 GL_(2)\mathrm{GL}_{2}GL2 or a certain unitary similitude group as in [33].
Problem 4.2. Prove a formula of the form
(4.1) tr ( f , p × Φ j [ IH ( Sh ¯ , Q l ¯ ) ] ) = e E e l l ( G ) ST e ( f e , , p f p e , ( j ) f e ) (4.1) tr ⁡ f ∞ , p × Φ j ∣ IH ⁡ Sh ¯ , Q l ¯ = ∑ e ∈ E e l l ( G )   ST e ⁡ f e , ∞ , p f p e , ( j ) f ∞ e {:(4.1)tr(f^(oo,p)xxPhi^(j)∣[IH( bar(Sh), bar(Q_(l)))])=sum_(e inE_(ell)(G))ST^(e)(f^(e,oo,p)f_(p)^(e,(j))f_(oo)^(e)):}\begin{equation*} \operatorname{tr}\left(f^{\infty, p} \times \Phi^{j} \mid\left[\operatorname{IH}\left(\overline{\operatorname{Sh}}, \overline{\mathbb{Q}_{l}}\right)\right]\right)=\sum_{e \in \mathcal{E}_{\mathrm{ell}}(G)} \operatorname{ST}^{e}\left(f^{e, \infty, p} f_{p}^{e,(j)} f_{\infty}^{e}\right) \tag{4.1} \end{equation*}(4.1)tr⁡(f∞,p×Φj∣[IH⁡(Sh¯,Ql¯)])=∑e∈Eell(G)STe⁡(fe,∞,pfpe,(j)f∞e)
where I H ( S h ¯ , Q ¯ l ) I H S h ¯ , Q ¯ l IH( bar(Sh), bar(Q)_(l))\mathrm{IH}\left(\overline{\mathrm{Sh}}, \overline{\mathbb{Q}}_{l}\right)IH(Sh¯,Q¯l) is the intersection cohomology of the Satake-Baily-Borel compactification of Sh (see [51, 3.4-3.5], for instance).
To obtain (4.1) from Theorem 3.3, one has to match the nonelliptic terms in S T e S T e ST^(e)\mathrm{ST}^{e}STe with the contribution to [ IH ( S h ¯ , Q ¯ l ) ] IH ⁡ S h ¯ , Q ¯ l [IH( bar(Sh), bar(Q)_(l))]\left[\operatorname{IH}\left(\overline{\mathrm{Sh}}, \overline{\mathbb{Q}}_{l}\right)\right][IH⁡(Sh¯,Q¯l)] from the boundaries. As a special case, if G / Z G G / Z G G//Z_(G)G / Z_{G}G/ZG is Q Q Q\mathbb{Q}Q anisotropic, then I H i ( S h ¯ , Q ¯ l ) = H c i ( S h , Q ¯ l ) I H i S h ¯ , Q ¯ l = H c i S h , Q ¯ l IH^(i)( bar(Sh), bar(Q)_(l))=H_(c)^(i)(Sh, bar(Q)_(l))\mathrm{IH}^{i}\left(\overline{\mathrm{Sh}}, \overline{\mathbb{Q}}_{l}\right)=H_{c}^{i}\left(\mathrm{Sh}, \overline{\mathbb{Q}}_{l}\right)IHi(Sh¯,Q¯l)=Hci(Sh,Q¯l) for each i 0 i ≥ 0 i >= 0i \geq 0i≥0, and the identity of Problem 4.1 should hold since there are no boundaries. In this sense, Problem 4.2 generalizes Problem 4.1. Problem 4.2 has been solved for Siegel modular varieties and certain unitary/orthogonal Shimura varieties by Morel and Zhu [ 50 , 52 , 73 ] [ 50 , 52 , 73 ] [50,52,73][50,52,73][50,52,73].

4.2. The global Langlands correspondence

The computation of â„“ â„“\ellâ„“-adic cohomology in Section 4.1 often leads to new instances of the global Langlands correspondence satisfying a local-global compatibility in the direction from automorphic representations to Galois representations, roughly stated as follows. Refer to Buzzard-Gee [7] for the definition of L L LLL-algebraicity and a full discussion of the conjecture, including a variant conjecture for C C CCC-algebraic representations.
Conjecture 4.3. Let F F FFF be a number field, and π = v π v Ï€ = ⊗ v ′ Ï€ v pi=ox_(v)^(')pi_(v)\pi=\otimes_{v}^{\prime} \pi_{v}Ï€=⊗v′πv an L-algebraic cuspidal automorphic representation of G ( A F ) G A F G(A_(F))G\left(\mathbb{A}_{F}\right)G(AF). Then for each prime â„“ â„“\ellâ„“ and each isomorphism ι : Q ¯ l C ι : Q ¯ l ≅ C iota: bar(Q)_(l)~=C\iota: \overline{\mathbb{Q}}_{l} \cong \mathbb{C}ι:Q¯l≅C, there exists a continuous representation ρ , l : Gal ( F ¯ / F ) L G ( Q ¯ l ) ρ â„“ , l : Gal ⁡ ( F ¯ / F ) → L G Q ¯ l rho_(â„“,l):Gal( bar(F)//F)rarr^(L)G( bar(Q)_(l))\rho_{\ell, l}: \operatorname{Gal}(\bar{F} / F) \rightarrow{ }^{L} G\left(\overline{\mathbb{Q}}_{l}\right)ρℓ,l:Gal⁡(F¯/F)→LG(Q¯l) such that the restriction of ρ , ι ρ â„“ , ι rho_(â„“,iota)\rho_{\ell, \iota}ρℓ,ι to Gal ( F ¯ v / F v ) Gal ⁡ F ¯ v / F v Gal( bar(F)_(v)//F_(v))\operatorname{Gal}\left(\bar{F}_{v} / F_{v}\right)Gal⁡(F¯v/Fv) is isomorphic to the unramified Langlands parameter of π v Ï€ v pi_(v)\pi_{v}Ï€v at almost all finite places v v vvv of F F FFF (where π v Ï€ v pi_(v)\pi_{v}Ï€v is unramified).
The relevance of Shimura varieties to the conjecture is as follows. A Shimura datum ( G , X ) ( G , X ) (G,X)(G, X)(G,X) determines a representation r X : L G G L ( V ) r X : L G → G L ( V ) r_(X):^(L)G rarrGL(V)r_{X}:{ }^{L} G \rightarrow \mathrm{GL}(V)rX:LG→GL(V) (up to isomorphism). Then one expects that the Galois representation r X ρ , L r X ∘ ρ â„“ , L r_(X)@rho_(â„“,L)r_{X} \circ \rho_{\ell, L}rX∘ρℓ,L is realized in the â„“ â„“\ellâ„“-adic cohomology of the associated Shimura varieties (more precisely, the π Ï€ ∞ pi^(oo)\pi^{\infty}π∞-isotypic part thereof), with several caveats including normalization issues (e.g., C C CCC-algebraic vs L L LLL-algebraic), Arthur packets, and endoscopic problems. These caveats often present much difficulty, and even if they are ignored, it is generally a subtle group-theoretic problem to recover ρ , ι ρ â„“ , ι rho_(â„“,iota)\rho_{\ell, \iota}ρℓ,ι from r ρ , l r ∘ ρ â„“ , l r@rho_(â„“,l)r \circ \rho_{\ell, l}r∘ρℓ,l for a set of representations r r rrr of L G L G ^(L)G{ }^{L} GLG. (Over global function fields, V. Lafforgue [38] solved the analogous problem in a revolutionary way via generalized pseudocharacters.)
The most fundamental case of Conjecture 4.3 is when G = G L n G = G L n G=GL_(n)G=\mathrm{GL}_{n}G=GLn. When F F FFF is a totally real or CM field and π Ï€ pi\piÏ€ satisfies a suitable self-duality condition, then the conjecture is proven in a series of papers making use of PEL-type Shimura varieties arising from a unitary similitude group by Clozel, Kottwitz, and others. (See [67] for a discussion and references.) The duality condition allows π Ï€ pi\piÏ€ to "descend" to an automorphic representation on the unitary similitude group as first observed by Clozel. The self-duality condition was later removed independently by Harris-Lan-Taylor-Thorne and Scholze [20,61], by exquisite p p ppp-adic congruences which are beyond the scope of this article.
The above results for G L n G L n GL_(n)\mathrm{GL}_{n}GLn imply new cases of Conjecture 4.3 (or a weaker form) for quasisplit unitary, symplectic, or special orthogonal groups G G GGG over a totally real or C M C M CM\mathrm{CM}CM field via twisted endoscopy by Arthur and Mok [1,49]. However, the conjecture for symplectic or orthogonal similitude groups does not follow easily. To get a feel for the difference, note that the dual groups of S p 2 n , S O 2 n S p 2 n , S O 2 n Sp_(2n),SO_(2n)\mathrm{Sp}_{2 n}, \mathrm{SO}_{2 n}Sp2n,SO2n are S O 2 n + 1 , S O 2 n S O 2 n + 1 , S O 2 n SO_(2n+1),SO_(2n)\mathrm{SO}_{2 n+1}, \mathrm{SO}_{2 n}SO2n+1,SO2n, which are embeddable in G L 2 n + 1 , G L 2 n G L 2 n + 1 , G L 2 n GL_(2n+1),GL_(2n)\mathrm{GL}_{2 n+1}, \mathrm{GL}_{2 n}GL2n+1,GL2n. In contrast, the dual groups of G S p 2 n G S p 2 n GSp_(2n)\mathrm{GSp}_{2 n}GSp2n and G S O 2 n G S O 2 n GSO_(2n)\mathrm{GSO}_{2 n}GSO2n are GSpin 2 n + 1 2 n + 1 2n+12 n+12n+1 and G S p i n 2 n G S p i n 2 n GSpin_(2n)\mathrm{GSpin}_{2 n}GSpin2n, whose faithful representations have dimensions at least 2 n 2 n 2^(n)2^{n}2n (achieved by the spin representation). In [35,37], Conjecture 4.3 is verified for G S p 2 n G S p 2 n GSp_(2n)\mathrm{GSp}_{2 n}GSp2n and a (possibly outer) form of G S O 2 n G S O 2 n GSO_(2n)\mathrm{GSO}_{2 n}GSO2n over a totally real field F F FFF under a simplifying hypothesis on π Ï€ pi\piÏ€. The basic input comes from Shimura varieties of abelian type associated with a form of Res F / Q G S p 2 n Res F / Q ⁡ G S p 2 n Res_(F//Q)GSp_(2n)\operatorname{Res}_{F / \mathbb{Q}} \mathrm{GSp}_{2 n}ResF/Q⁡GSp2n, resp. Res F / Q G S O 2 n Res F / Q ⁡ G S O 2 n Res_(F//Q)GSO_(2n)\operatorname{Res}_{F / \mathbb{Q}} \mathrm{GSO}_{2 n}ResF/Q⁡GSO2n, where r X r X r_(X)r_{X}rX is essentially the spin representation, resp. a half-spin representation. Both problems in Section 4.1 have positive answers in that setup.

5. SHIMURA VARIETIES WITH BAD REDUCTION, PART I

Let ( G , X ) ( G , X ) (G,X)(G, X)(G,X) be a Shimura datum. At each prime p p ppp such that ( G , X ) ( G , X ) (G,X)(G, X)(G,X) can be promoted to an unramified Shimura datum ( G , X , p , E ) ( G , X , p , E ) (G,X,p,E)(G, X, p, \mathscr{E})(G,X,p,E), we discuss three methods to study the cohomology H c ( S h , Q ¯ l ) H c S h , Q ¯ l H_(c)(Sh, bar(Q)_(l))H_{c}\left(\mathrm{Sh}, \overline{\mathbb{Q}}_{l}\right)Hc(Sh,Q¯l) as a G ( A ) × Gal ( E ¯ p / E p ) G A ∞ × Gal ⁡ E ¯ p / E p G(A^(oo))xx Gal( bar(E)_(p)//E_(p))G\left(\mathbb{A}^{\infty}\right) \times \operatorname{Gal}\left(\bar{E}_{\mathfrak{p}} / E_{\mathfrak{p}}\right)G(A∞)×Gal⁡(E¯p/Ep)-module at each prime p p p\mathfrak{p}p of E E EEE over p p ppp. 3 3 ^(3){ }^{3}3 The "bad reduction" in the section title means that the level subgroup K p K p = E ( Z p ) K p ′ ⊂ K p = E Z p K_(p)^(')subK_(p)=E(Z_(p))K_{p}^{\prime} \subset K_{p}=\mathscr{E}\left(\mathbb{Z}_{p}\right)Kp′⊂Kp=E(Zp) at p p ppp is allowed to be arbitrarily small, in which case integral models typically have bad reduction mod p p ppp. The complicated geometry may be understood better through stratifications.
There are several stratifications of interest on Shimura varieties (cf. [22]) but the most relevant to us is the Newton stratification. In the Hodge-type case, this yields a partition of the mod p mod p mod p\bmod pmodp Shimura variety into finitely many locally closed subsets, which can be equipped with the reduced subscheme structure, according to the isogeny class of p p ppp-divisible groups with additional structure. The unique closed stratum is called the basic stratum and corresponds to the p p ppp-divisible group that is "most supersingular" under the given constraint.
The first method is a p p ppp-adic uniformization of Shimura varieties as pioneered by Čerednik and Drinfeld, and further developed by Rapoport-Zink, Fargues, Kim, and Howard-Pappas [11,23,26,57]. Let S h K p K p basic S h K p K p ′ basic  Sh_(K^(p)K_(p)^('))^("basic ")\mathrm{Sh}_{K^{p} K_{p}^{\prime}}^{\text {basic }}ShKpKp′basic  denote the basic locus in the rigid analytification of S h K p K p S h K p K p ′ Sh_(K^(p)K_(p)^('))\mathrm{Sh}_{K^{p} K_{p}^{\prime}}ShKpKp′ over E p E p E_(p)E_{\mathfrak{p}}Ep, defined to be the preimage of the basic stratum under the specialization map towards the special fiber. The fundamental result asserts that S h K p K p basi S h K p K p ′ basi  Sh_(K^(p)K_(p)^('))^("basi ")\mathrm{Sh}_{K^{p} K_{p}^{\prime}}^{\text {basi }}ShKpKp′basi  is uniformized by the Rapoport-Zink space with level K p K p ′ K_(p)^(')K_{p}^{\prime}Kp′ arising from the corresponding basic isogeny class. A prominent application is to prove new cases of the Kottwitz conjecture on the cohomology of basic Rapoport-Zink spaces and their generalizations [11,19,25]. Hansen's work [19] points to a synergy between the global method here and Fargues-Scholze's purely local geometric construction of the local Langlands correspondence [12].
Next we discuss the Harris-Taylor method [21, cHAPS. Iv-v] based on a product structure, namely coverings of Newton strata by the products of Igusa varieties and RapoportZink spaces. The outcome is known as Mantovan's formula [44] (generalizing [21, cHAP. Iv]), which expresses the cohomology of Newton strata in terms of that of Igusa varieties and Rapoport-Zink spaces. In the basic case, this is closely related to the p p ppp-adic uniformization. Hamacher-Kim [18] extended Mantovan's formula and the product structure to Hodge-type Shimura varieties.
To go further, it is desirable to understand the cohomology of Igusa varieties - we address this problem in Section 7 below. Granting this, and putting different Newton strata together, we have a formula relating [ H c ( S h , Q ¯ l ) ] H c S h , Q ¯ l [H_(c)(Sh, bar(Q)_(l))]\left[H_{c}\left(\mathrm{Sh}, \overline{\mathbb{Q}}_{l}\right)\right][Hc(Sh,Q¯l)] to the cohomology of Rapoport-Zink spaces. Then our knowledge about [ H c ( S h , Q ¯ l ) ] H c S h , Q ¯ l [H_(c)(Sh, bar(Q)_(l))]\left[H_{c}\left(\mathrm{Sh}, \overline{\mathbb{Q}}_{l}\right)\right][Hc(Sh,Q¯l)] tells us something nontrivial about the cohomology of Rapoport-Zink spaces, and vice versa. This observation turned out to be useful for proving local-global compatibility, i.e., identifying the local Galois action for the Galois representations in Conjecture 4.3 at ramified primes (see [21, cHAP. vII], [65]) and also for understanding the cohomology of basic/nonbasic Rapoport-Zink spaces [2-4,66].
Last but not least, there is Scholze's extension of the Langlands-Kottwitz approach from the hyperspecial level at p p ppp to arbitrarily small level subgroup at p p ppp. (See Section 6.2 below for another generalization.) One seeks for the following analogue of (3.7), where τ Ï„ ∈ tau in\tau \inτ∈ W E p W E p W_(E_(p))W_{E_{\mathfrak{p}}}WEp is a Weil group element with positive valuation, h H ( G ( Q p ) ) h ∈ H G Q p h inH(G(Q_(p)))h \in \mathscr{H}\left(G\left(\mathbb{Q}_{p}\right)\right)h∈H(G(Qp)) has support contained in E ( Z p ) E Z p E(Z_(p))\mathcal{E}\left(\mathbb{Z}_{p}\right)E(Zp), and the sum is over the same set of c c ccc :
(5.1) tr ( f , p × h × τ [ H c ( S h , Q ¯ l ) ] ) = c c ( c ) O γ ( c ) ( f , p ) T O δ ( c ) ( ϕ τ , h ) (5.1) tr ⁡ f ∞ , p × h × Ï„ ∣ H c S h , Q ¯ l = ∑ c   c ( c ) O γ ( c ) f ∞ , p T O δ ( c ) Ï• Ï„ , h {:(5.1)tr(f^(oo,p)xx h xx tau∣[H_(c)(Sh, bar(Q)_(l))])=sum_(c)c(c)O_(gamma(c))(f^(oo,p))TO_(delta(c))(phi_(tau,h)):}\begin{equation*} \operatorname{tr}\left(f^{\infty, p} \times h \times \tau \mid\left[H_{c}\left(\mathrm{Sh}, \overline{\mathbb{Q}}_{l}\right)\right]\right)=\sum_{\mathrm{c}} c(\mathrm{c}) O_{\gamma(\mathfrak{c})}\left(f^{\infty, p}\right) T O_{\delta(\mathfrak{c})}\left(\phi_{\tau, h}\right) \tag{5.1} \end{equation*}(5.1)tr⁡(f∞,p×h×τ∣[Hc(Sh,Q¯l)])=∑cc(c)Oγ(c)(f∞,p)TOδ(c)(ϕτ,h)
This has been verified by Scholze [59] for PEL-type data and by Youcis [71] for abeliantype data. As an application, Scholze gave a new proof and characterization of the local Langlands correspondence for G L n G L n GL_(n)\mathrm{GL}_{n}GLn over p p ppp-adic fields [60] via a base-change transfer of ϕ τ , h Ï• Ï„ , h phi_(tau,h)\phi_{\tau, h}ϕτ,h. A generalization of the latter to other groups was conjectured in [62] and partially proved for unitary groups by Bertoloni Meli and Youcis [5].
In the proof of (5.1), one can push-forward from arbitrarily small level K p K p ′ K_(p)^(')K_{p}^{\prime}Kp′ down to hyperspecial level K p K p K_(p)K_{p}Kp at the expense of complicating the coefficient sheaf. Applying the fixed-point formula to this, one can exploit knowledge of the fixed-points (coming from results on the LR conjecture). The main problem is to identify the local terms, which are shown to be encoded by a locally constant compactly supported function ϕ τ , h Ï• Ï„ , h phi_(tau,h)\phi_{\tau, h}ϕτ,h at p p ppp constructed from deformation spaces of p p ppp-divisible groups with additional structures.

6. SHIMURA VARIETIES WITH BAD REDUCTION, PART II

Let ( G , X ) ( G , X ) (G,X)(G, X)(G,X) be a Shimura datum, and p p ppp a prime. In this section, we survey generalizations of Sections 2 5 2 − 5 2-52-52−5 in the setup where G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp is allowed to be ramified (thus there may be no unramified Shimura datum of the form ( G , X , p , E ) ( G , X , p , E ) (G,X,p,E)(G, X, p, \mathcal{E})(G,X,p,E) ). We recommend the articles [ 14 , 53 , 56 ] [ 14 , 53 , 56 ] [14,53,56][14,53,56][14,53,56] for introductions to the contents of this section.

6.1. The LR conjecture in the parahoric case

From now until the end of Section 6, assume that ( G , X ) ( G , X ) (G,X)(G, X)(G,X) is of abelian type. We fix K p G ( Q p ) K p ⊂ G Q p K_(p)sub G(Q_(p))K_{p} \subset G\left(\mathbb{Q}_{p}\right)Kp⊂G(Qp) a parahoric subgroup, and p p p\mathfrak{p}p a place of E E EEE over p p ppp. In this setting, KisinPappas [30] constructed an integral model S K p S K p S_(K_(p))\mathscr{S}_{K_{p}}SKp over O E p O E p O_(E_(p))\mathcal{O}_{E_{\mathfrak{p}}}OEp under a mild hypothesis, which are canonical in the sense of [54].
With the integral model as above, we can state versions of the Langlands-Rapoport conjecture analogous to Conjecture 3.1, cf. [56, §9]. 4 4 ^(4){ }^{4}4 One can extend the notion of isogeny classes on S K p ( F ¯ p ) S K p F ¯ p S_(K_(p))( bar(F)_(p))\mathscr{S}_{K_{p}}\left(\overline{\mathbb{F}}_{p}\right)SKp(F¯p) and admissible morphisms ϕ : Q F G Ï• : Q → F G phi:QrarrF_(G)\phi: \mathbb{Q} \rightarrow \mathbb{F}_{G}Ï•:Q→FG to the parahoric setup, following [30] and [56], respectively. Thus we can consider the set I I I\mathbb{I}I of isogeny classes and the set J J J\mathbb{J}J of conjugacy classes of admissible morphisms. The set S ( χ ) S ( χ ) S(chi)S(\mathscr{\chi})S(χ) of F ¯ p F ¯ p bar(F)_(p)\overline{\mathbb{F}}_{p}F¯p-points in each isogeny class I I I ∈ I IinI\mathscr{I} \in \mathbb{I}I∈I is still described by (3.4), with X p ( ) X p ( â„“ ) X_(p)(â„“)X_{p}(\mathcal{\ell})Xp(â„“) a suitable affine Deligne-Lusztig variety at the parahoric level K p K p K_(p)K_{p}Kp. Analogously we define S τ ( ϕ ) S Ï„ ( Ï• ) S_(tau)(phi)S_{\tau}(\phi)SÏ„(Ï•) and S τ ( L ) S Ï„ ( L ) S_(tau)(L)S_{\tau}(\mathcal{L})SÏ„(L) for each admissible ϕ Ï• phi\phiÏ• and J J J ∈ J JinJ\mathcal{J} \in \mathbb{J}J∈J, with X p ( ϕ ) X p ( Ï• ) X_(p)(phi)X_{p}(\phi)Xp(Ï•) in (3.5) also adapted to the parahoric level K p K p K_(p)K_{p}Kp.
Conjecture 6.1. With the above definitions, there exists a G ( A , p ) × Φ Z G A ∞ , p × Φ Z G(A^(oo,p))xxPhi^(Z)G\left(\mathbb{A}^{\infty, p}\right) \times \Phi^{\mathbb{Z}}G(A∞,p)×ΦZ-equivariant bijection S K p ( F ¯ p ) g J S τ ( J ) ( J ) S K p F ¯ p ≅ ⨆ g ∈ J   S Ï„ ( J ) ( J ) S_(K_(p))( bar(F)_(p))~=⨆_(ginJ)S_(tau(J))(J)\mathscr{S}_{K_{p}}\left(\overline{\mathbb{F}}_{p}\right) \cong \bigsqcup_{\mathcal{g} \in \mathbb{J}} S_{\tau(\mathcal{J})}(\mathcal{J})SKp(F¯p)≅⨆g∈JSÏ„(J)(J) such that the exact analogue of ( L R 1 ) L R 1 (LR_(1))\left(\mathrm{LR}_{1}\right)(LR1), resp. ( L R 0 ) L R 0 (LR_(0))\left(\mathrm{LR}_{0}\right)(LR0) and (LR), holds true.
Van Hoften and Zhou [68,72] proved the following theorem.
Theorem 6.2. Statement ( L R 1 ) L R 1 (LR_(1))\left(\mathrm{LR}_{1}\right)(LR1) of Conjecture 6.1 is true if G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp is quasisplit, under a mild technical hypothesis.
The stronger statement ( L R 0 ) L R 0 (LR_(0))\left(\mathrm{LR}_{0}\right)(LR0) is expected to be within reach under the same hypothesis, by extending the argument from [31] to the parahoric setting.
To prove Theorem 6.2, the essential case is when ( G , X ) ( G , X ) (G,X)(G, X)(G,X) is of Hodge type. Zhou proves the very special parahoric case of the conjecture. Van Hoften deduces the case of general parahoric subgroup K p K p ′ K_(p)^(')K_{p}^{\prime}Kp′, contained in a very special parahoric K p K p K_(p)K_{p}Kp, by studying the localization maps from Shimura varieties of level K p K p ′ K_(p)^(')K_{p}^{\prime}Kp′ and K p K p K_(p)K_{p}Kp, to their respective moduli spaces of local Shtukas. The maps are roughly given by assigning to each abelian variety the associated p p ppp-divisible group in terms of the moduli problems. Via the forgetful maps from level K p K p ′ K_(p)^(')K_{p}^{\prime}Kp′ down to level K p K p K_(p)K_{p}Kp, one can form a commutative square diagram. The central claim is that the diagram is Cartesian, from which the LR conjecture at level K p K p ′ K_(p)^(')K_{p}^{\prime}Kp′ can be deduced from the known case at level K p K p K_(p)K_{p}Kp. The proof of the claim eventually rests on understanding the irreducible components of Kottwitz-Rapoport strata in the situation of the diagram.

6.2. Semisimple zeta functions and Haines-Kottwitz's test function conjecture

At primes of bad reduction, it is useful to compute the semisimple local zeta factor at p p ppp instead of the (true) local factor of the Hasse-Weil zeta function (of Shimura varieties) as the former is more amenable to computation. The latter can be recovered from the former in the cases where the weight-monodromy conjecture is known [55].
4 One can remove the assumption in [56] that G der G der  G^("der ")G^{\text {der }}Gder  is simply connected, by adopting Kisin's formulation in [29] via strictly monoidal categories.
Just like the local zeta factor at p p ppp can be computed in terms of the trace of powers of Frobenius on the Frobenius-invariant subspace of the cohomology with compact support, the semisimple local factor can be described in terms of the trace of powers of Frobenius on the derived Frobenius-invariants; such a trace is called the semisimple trace and will be denoted by tr s s tr s s tr^(ss)\operatorname{tr}^{\mathrm{ss}}trss (see [ 55 , $ 2 ] , [ 15 , $ 3.1 ] [ 55 , $ 2 ] , [ 15 , $ 3.1 ] [55,$2],[15,$3.1][55, \$ 2],[15, \$ 3.1][55,$2],[15,$3.1] ). Thus a key is to establish the following generalization of (3.7), which recovers (3.7) if K p K p K_(p)K_{p}Kp is hyperspecial, due to Haines and Kottwitz [14, 86.1]. The summation is over the same set as in (3.7).
Conjecture 6.3. Let f , p H ( G ( A , p ) ) f ∞ , p ∈ H G A ∞ , p f^(oo,p)inH(G(A^(oo,p)))f^{\infty, p} \in \mathscr{H}\left(G\left(\mathbb{A}^{\infty, p}\right)\right)f∞,p∈H(G(A∞,p)). For all sufficiently large integers j 1 j ≫ 1 j≫1j \gg 1j≫1, there exist test functions ϕ H K ( j ) H ( G ( Q p j r ) ) Ï• H K ( j ) ∈ H G Q p j r phi_(HK)^((j))inH(G(Q_(p)jr))\phi_{\mathrm{HK}}^{(j)} \in \mathscr{H}\left(G\left(\mathbb{Q}_{p} j r\right)\right)Ï•HK(j)∈H(G(Qpjr)) such that
(6.1) tr s s ( f , p × Φ p j [ H c ( Sh K p , Q ¯ l ) ] ) = c c ( c ) O γ ( c ) ( f , p ) T O δ ( c ) ( ϕ H K ( j ) ) (6.1) tr s s ⁡ f ∞ , p × Φ p j ∣ H c Sh K p , Q ¯ l = ∑ c   c ( c ) O γ ( c ) f ∞ , p T O δ ( c ) Ï• H K ( j ) {:(6.1)tr^(ss)(f^(oo,p)xxPhi_(p)^(j)∣[H_(c)(Sh_(K_(p)), bar(Q)_(l))])=sum_(c)c(c)O_(gamma(c))(f^(oo,p))TO_(delta(c))(phi_(HK)^((j))):}\begin{equation*} \operatorname{tr}^{\mathrm{ss}}\left(f^{\infty, p} \times \Phi_{\mathfrak{p}}^{j} \mid\left[H_{c}\left(\operatorname{Sh}_{K_{p}}, \overline{\mathbb{Q}}_{l}\right)\right]\right)=\sum_{\mathrm{c}} c(\mathrm{c}) O_{\gamma(\mathrm{c})}\left(f^{\infty, p}\right) T O_{\delta(\mathrm{c})}\left(\phi_{\mathrm{HK}}^{(j)}\right) \tag{6.1} \end{equation*}(6.1)trss⁡(f∞,p×Φpj∣[Hc(ShKp,Q¯l)])=∑cc(c)Oγ(c)(f∞,p)TOδ(c)(Ï•HK(j))
Moreover, ϕ H K ( j ) Ï• H K ( j ) phi_(HK)^((j))\phi_{\mathrm{HK}}^{(j)}Ï•HK(j) may be given by an explicit recipe only in terms of local data at p p ppp.
When K p K p K_(p)K_{p}Kp is a parahoric subgroup of G ( Q p ) G Q p G(Q_(p))G\left(\mathbb{Q}_{p}\right)G(Qp), one can be more concrete about ϕ H K ( j ) Ï• H K ( j ) phi_(HK)^((j))\phi_{\mathrm{HK}}^{(j)}Ï•HK(j) following [14, 87]: ϕ H K ( j ) Ï• H K ( j ) phi_(HK)^((j))\phi_{\mathrm{HK}}^{(j)}Ï•HK(j) should admit a geometric construction via nearby cycles on local models, as well as a representation-theoretic description in terms of the Langlands correspondence. That the two descriptions for ϕ H K ( j ) Ï• H K ( j ) phi_(HK)^((j))\phi_{\mathrm{HK}}^{(j)}Ï•HK(j) coincide is the test function conjecture verified by Haines-Richarz [16,17] under a very mild hypothesis. (See [14, §8] for prior and related results.) The proof is based on geometry of mixed-characteristic affine Grassmanians and the geometric Satake equivalence. It remains to combine their theorem with the results in Section 6.1 to obtain new cases of Conjecture 6.3 and its stabilized form, so as to determine the semisimple zeta factor at p p ppp. This requires an endoscopic understanding of ϕ H K ( j ) Ï• H K ( j ) phi_(HK)^((j))\phi_{\mathrm{HK}}^{(j)}Ï•HK(j), cf. [14, §6.2]; a simple exemplary case is demonstrated in [ 14 , $ 6.3 ] [ 14 , $ 6.3 ] [14,$6.3][14, \$ 6.3][14,$6.3], where endoscopic problems disappear.
In a related but somewhat different direction (cf. the last two paragraphs in [14, §8.4]), the Langlands-Kottwitz-Scholze approach discussed in Section 5 should extend to the current setup despite the absence of hyperspecial subgroups at p p ppp, at least when the results of Section 6.1 are available for some parahoric subgroups.
Problem 6.4. Prove the analogue of (5.1) for general Shimura data ( G , X ) G , X ) G,X)G, X)G,X) and primes p p ppp.

7. IGUSA VARIETIES

Igusa curves were introduced to understand the geometry of modular curves modulo p p ppp when the level is divisible by a prime p p ppp [24]. The construction has been generalized by Harris-Taylor [21] and Mantovan [44] in the PEL-type case, and most recently to the setup of Kisin-Pappas models for Hodge-type Shimura varieties by Hamacher-Kim [18]. Igusa varieties have a variety of applications to p p ppp-adic and mod p p ppp modular forms, cohomology of Shimura varieties, the Langlands correspondence, and some more. We refer to the introduction of [36] for further details and references. In this section, we concentrate on computing the â„“ â„“\ellâ„“-adic cohomology of Igusa varieties via an analogue of the LR conjecture.

7.1. The LR conjecture for Igusa varieties

Let ( G , X , p , E ) ( G , X , p , E ) (G,X,p,E)(G, X, p, \mathcal{E})(G,X,p,E) be an unramified Shimura datum of Hodge type, with a fixed embedding of ( G , X ) ( G , X ) (G,X)(G, X)(G,X) into a Siegel Shimura datum. Put K p := E ( Z p ) K p := E Z p K_(p):=E(Z_(p))K_{p}:=\mathscr{E}\left(\mathbb{Z}_{p}\right)Kp:=E(Zp), a hyperspecial subgroup of G ( Q p ) G Q p G(Q_(p))G\left(\mathbb{Q}_{p}\right)G(Qp). Let A A A\mathscr{A}A denote the abelian scheme over S K p S K p S_(K_(p))\mathscr{S}_{K_{p}}SKp pulled back from the universal abelian scheme over the ambient Siegel moduli scheme. Thus we have a p p ppp-divisible group A [ p ] A p ∞ A[p^(oo)]\mathcal{A}\left[p^{\infty}\right]A[p∞] over S K p S K p S_(K_(p))\mathscr{S}_{K_{p}}SKp with G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp-structure in some precise sense.
Now fix a p p ppp-divisible group Σ Î£ Sigma\SigmaΣ over F ¯ p F ¯ p bar(F)_(p)\overline{\mathbb{F}}_{p}F¯p with G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp-structure. By Dieudonné theory, this determines b G ( Q ˘ p ) b ∈ G Q ˘ p b in G(Q^(˘)_(p))b \in G\left(\breve{\mathbb{Q}}_{p}\right)b∈G(Q˘p) (up to replacing b b bbb with g 1 b x σ ( g ) g − 1 b x σ ( g ) g^(-1)b_(x)sigma(g)g^{-1} b_{x} \sigma(g)g−1bxσ(g) for g E ( Z ˘ p ) g ∈ E Z ˘ p g inE(Z^(˘)_(p))g \in \mathcal{E}\left(\breve{\mathbb{Z}}_{p}\right)g∈E(Z˘p) ). Write J b ( Q p ) J b Q p J_(b)(Q_(p))J_{b}\left(\mathbb{Q}_{p}\right)Jb(Qp) for the group of self-quasi-isogenies of Σ Î£ Sigma\SigmaΣ respecting the G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp-structure. As a Q p Q p Q_(p)\mathbb{Q}_{p}Qp-algebraic group, J b J b J_(b)J_{b}Jb is known to be an inner form of a Levi subgroup of G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp. We define Ig b Ig b Ig_(b)\operatorname{Ig}_{b}Igb to be the parameter space (in the category of perfect F ¯ p F ¯ p bar(F)_(p)\overline{\mathbb{F}}_{p}F¯p-schemes) of G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp-structure-preserving isomorphisms between A [ p ] A p ∞ A[p^(oo)]\mathcal{A}\left[p^{\infty}\right]A[p∞] and (the constant family of) Σ Î£ Sigma\SigmaΣ over S K p , F ¯ p S K p , F ¯ p S_(K_(p), bar(F)_(p))\mathscr{S}_{K_{p}, \overline{\mathbb{F}}_{p}}SKp,F¯p. The scheme Ig b Ig b Ig_(b)\operatorname{Ig}_{b}Igb is nonempty if and only if the image of b b bbb in Kottwitz's set B ( G ) B ( G ) B(G)B(G)B(G) lies in the finite subset B ( G , μ X 1 ) B G , μ X − 1 B(G,mu_(X)^(-1))B\left(G, \mu_{X}^{-1}\right)B(G,μX−1), where μ X μ X mu_(X)\mu_{X}μX is the Hodge cocharacter of G G GGG determined by ( G , X ) ( G , X ) (G,X)(G, X)(G,X) up to conjugacy. (The same set B ( G , μ X 1 ) B G , μ X − 1 B(G,mu_(X)^(-1))B\left(G, \mu_{X}^{-1}\right)B(G,μX−1) labels the Newton strata, cf. Section 5.) There is a natural action of G ( A , p ) × J b ( Q p ) G A ∞ , p × J b Q p G(A^(oo,p))xxJ_(b)(Q_(p))G\left(\mathbb{A}^{\infty, p}\right) \times J_{b}\left(\mathbb{Q}_{p}\right)G(A∞,p)×Jb(Qp) on Ig b Ig b Ig_(b)\operatorname{Ig}_{b}Igb, thus also on its â„“ â„“\ellâ„“-adic cohomology.
We may and will replace Σ Î£ Sigma\SigmaΣ with an isogenous p p ppp-divisible group which is completely slope divisible, since the isomorphism class of I g b I g b Ig_(b)\mathrm{Ig}_{b}Igb with the group action is invariant under such a replacement. The advantage of doing so is that I g b I g b Ig_(b)\mathrm{Ig}_{b}Igb can be written as the projective limit of finite-type varieties (up to taking perfection, which does not affect cohomology) by trivializing only a finite p p ppp-power torsion subgroup and by fixing a level subgroup K p K p ⊂ K^(p)subK^{p} \subsetKp⊂ G ( A , p ) G A ∞ , p G(A^(oo,p))G\left(\mathbb{A}^{\infty, p}\right)G(A∞,p) away from p p ppp at a time. With some care, the projective system of varieties can be defined over a common finite field. This enables us to apply the Fujiwara-Varshavsky fixed-point formula to compute the cohomology (with compact support) at each finite level, provided that we understand the structure of Ig b ( F ¯ p ) Ig b ⁡ F ¯ p Ig_(b)( bar(F)_(p))\operatorname{Ig}_{b}\left(\overline{\mathbb{F}}_{p}\right)Igb⁡(F¯p). Thus we are prompted to think about the analogue of the LR conjecture for Igusa varieties.
In analogy with (3.3) and (3.4), keeping the same definition of I I I\mathbb{I}I and J J J\mathbb{J}J, we have the partition S Ig b ( F ¯ p ) = I S I g b ( ) S Ig b F ¯ p = ∐ â„“ ∈ I   S I g b ( â„“ ) S^(Ig_(b))( bar(F)_(p))=∐_(â„“inI)S^(Ig_(b))(â„“)S^{\operatorname{Ig}_{b}}\left(\overline{\mathbb{F}}_{p}\right)=\coprod_{\ell \in \mathbb{I}} S^{\mathrm{Ig}_{b}}(\mathcal{\ell})SIgb(F¯p)=∐ℓ∈ISIgb(â„“) according to isogeny classes of abelian varieties, with
(7.1) S Ig b ( l ) I l ( Q ) ( X p Ig b ( l ) × X p ( l ) ) (7.1) S Ig b ( l ) ≅ I l ( Q ) ∖ X p Ig b ( l ) × X p ( l ) {:(7.1)S^(Ig_(b))(l)~=I_(l)(Q)\\(X_(p)^(Ig_(b))(l)xxX^(p)(l)):}\begin{equation*} S^{\operatorname{Ig}_{b}}(\mathcal{l}) \cong I_{\mathscr{l}}(\mathbb{Q}) \backslash\left(X_{p}^{\operatorname{Ig}_{b}}(\mathcal{l}) \times X^{p}(\mathcal{l})\right) \tag{7.1} \end{equation*}(7.1)SIgb(l)≅Il(Q)∖(XpIgb(l)×Xp(l))
The G ( A , p ) G A ∞ , p G(A^(oo,p))G\left(\mathbb{A}^{\infty, p}\right)G(A∞,p)-set X p ( d ) X p ( d ) X^(p)(d)X^{p}(\mathscr{d})Xp(d) is the same as before, but the difference from Section 3 is that X p I g ( d ) X p I g ( d ) X_(p)^(Ig)(d)X_{p}^{\mathrm{Ig}}(\mathcal{d})XpIg(d) is no longer an affine Deligne-Lusztig variety but a right J b ( Q p ) J b Q p J_(b)(Q_(p))J_{b}\left(\mathbb{Q}_{p}\right)Jb(Qp)-torsor. Turning to the other side of the L R L R LR\mathrm{LR}LR conjecture, let J J J ∈ J JinJ\mathscr{J} \in \mathbb{J}J∈J. It is natural to impose the so-called b b bbb admissibility condition on J J J\mathcal{J}J at p p ppp, which is the group-theoretic analogue of the condition that a p p ppp-divisible group is isogenous to Σ Î£ Sigma\SigmaΣ (with G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp-structure). For b b bbb-admissible L L L\mathcal{L}L, we set
S τ Ig b ( J ) := I ϕ ( Q ) τ ( X p I g b ( J ) × X p ( d ) ) S Ï„ Ig b ( J ) := I Ï• ( Q ) ∖ Ï„ X p I g b ( J ) × X p ( d ) S_(tau)^(Ig_(b))(J):=I_(phi)(Q)\\_(tau)(X_(p)^(Ig_(b))(J)xxX^(p)(d))S_{\tau}^{\operatorname{Ig}_{b}}(\mathcal{J}):=I_{\phi}(\mathbb{Q}) \backslash_{\tau}\left(X_{p}^{\mathrm{Ig}_{b}}(\mathcal{J}) \times X^{p}(\mathcal{d})\right)SÏ„Igb(J):=IÏ•(Q)∖τ(XpIgb(J)×Xp(d))
with the same X p ( J ) X p ( J ) X^(p)(J)X^{p}(\mathcal{J})Xp(J) as in Section 3, and a suitably defined right J b ( Q p ) J b Q p J_(b)(Q_(p))J_{b}\left(\mathbb{Q}_{p}\right)Jb(Qp)-torsor X p I g b ( L ) X p I g b ( L ) X_(p)^(Ig_(b))(L)X_{p}^{\mathrm{Ig}_{b}}(\mathcal{L})XpIgb(L), where the τ Ï„ tau\tauÏ„-twisted quotient can be defined again as in Section 3. We are ready to state versions of the LR conjecture for Igusa varieties in parallel with Conjecture 3.1, for unramified Shimura data ( G , X , p , E ) ( G , X , p , E ) (G,X,p,E)(G, X, p, \mathcal{E})(G,X,p,E) of Hodge type.
Conjecture 7.1. There is a bijection of right J b ( Q p ) × G ( A , p ) J b Q p × G A ∞ , p J_(b)(Q_(p))xx G(A^(oo,p))J_{b}\left(\mathbb{Q}_{p}\right) \times G\left(\mathbb{A}^{\infty, p}\right)Jb(Qp)×G(A∞,p)-sets
Ig b ( F ¯ p ) L J b -ddm. S τ ( f ) I g b ( J ) Ig b ⁡ F ¯ p ≃ ∐ L ∈ J b -ddm.    S Ï„ ( f ) I g b ( J ) Ig_(b)( bar(F)_(p))≃∐_({:[LinJ],[b"-ddm. "]:})S_(tau(f))^(Ig_(b))(J)\operatorname{Ig}_{b}\left(\overline{\mathbb{F}}_{p}\right) \simeq \coprod_{\substack{\mathcal{L} \in \mathbb{J} \\ b \text {-ddm. }}} S_{\tau(\mathcal{f})}^{\mathrm{Ig}_{b}}(\mathcal{J})Igb⁡(F¯p)≃∐L∈Jb-ddm. SÏ„(f)Igb(J)
where { τ ( d ) } { Ï„ ( d ) } {tau(d)}\{\tau(\mathcal{d})\}{Ï„(d)} over the set of b b bbb-admissible I I I\mathcal{I}I satisfies the conditions in ( L R 1 ) L R 1 (LR_(1))\left(\mathrm{LR}_{1}\right)(LR1), resp. ( L R 0 ) L R 0 (LR_(0))\left(\mathrm{LR}_{0}\right)(LR0) and (LR).
Mack-Crane proved the following theorem in his thesis [43], where c , c ( c ) c , c ( c ) c,c(c)c, c(c)c,c(c), and γ ( c ) γ ( c ) gamma(c)\gamma(c)γ(c) are the same as in Section 3, but we impose a b b bbb-admissibility condition on the Kottwitz parameter c c c\mathrm{c}c inherited from the similar condition on L L L\mathcal{L}L, and each c c c\mathrm{c}c gives rise to a conjugacy class of δ ( c ) δ ′ ( c ) delta^(')(c)\delta^{\prime}(c)δ′(c) in J b ( Q p ) J b Q p J_(b)(Q_(p))J_{b}\left(\mathbb{Q}_{p}\right)Jb(Qp), along which we compute the (untwisted) orbital integral O δ ( c ) ( ϕ p ) O δ ′ ( c ) Ï• p ′ O_(delta^(')(c))(phi_(p)^('))O_{\delta^{\prime}(c)}\left(\phi_{p}^{\prime}\right)Oδ′(c)(Ï•p′).
Theorem 7.2. The ( L R 0 ) L R 0 (LR_(0))\left(\mathrm{LR}_{0}\right)(LR0)-version of Conjecture 7.1 is true. Moreover, the following analogue of (3.7) holds true for f , p H ( G ( A , p ) ) f ∞ , p ∈ H G A ∞ , p f^(oo,p)inH(G(A^(oo,p)))f^{\infty, p} \in \mathscr{H}\left(G\left(\mathbb{A}^{\infty, p}\right)\right)f∞,p∈H(G(A∞,p)) and sufficiently many functions ϕ p Ï• p ′ ∈ phi_(p)^(')in\phi_{p}^{\prime} \inÏ•p′∈ H ( J b ( Q p ) ) : H J b Q p : H(J_(b)(Q_(p))):\mathscr{H}\left(J_{b}\left(\mathbb{Q}_{p}\right)\right):H(Jb(Qp)):
(7.2) tr ( f , p × ϕ p [ H c ( I g b , Q ¯ l ) ] ) = c : b -adm. c ( c ) O γ ( c ) ( f , p ) O δ ( c ) ( ϕ p ) (7.2) tr ⁡ f ∞ , p × Ï• p ′ ∣ H c I g b , Q ¯ l = ∑ c : b -adm.    c ( c ) O γ ( c ) f ∞ , p O δ ′ ( c ) Ï• p ′ {:(7.2)tr(f^(oo,p)xxphi_(p)^(')∣[H_(c)(Ig_(b), bar(Q)_(l))])=sum_(c:b"-adm. ")c(c)O_(gamma(c))(f^(oo,p))O_(delta^(')(c))(phi_(p)^(')):}\begin{equation*} \operatorname{tr}\left(f^{\infty, p} \times \phi_{p}^{\prime} \mid\left[H_{c}\left(\mathrm{Ig}_{b}, \overline{\mathbb{Q}}_{l}\right)\right]\right)=\sum_{\mathrm{c}: b \text {-adm. }} c(c) O_{\gamma(\mathrm{c})}\left(f^{\infty, p}\right) O_{\delta^{\prime}(c)}\left(\phi_{p}^{\prime}\right) \tag{7.2} \end{equation*}(7.2)tr⁡(f∞,p×ϕp′∣[Hc(Igb,Q¯l)])=∑c:b-adm. c(c)Oγ(c)(f∞,p)Oδ′(c)(Ï•p′)
By "sufficiently many," we mean that the traces for such a set of functions are enough to determine [ H c ( Ig b , Q ¯ l ) ] H c Ig b , Q ¯ l [H_(c)(Ig_(b), bar(Q)_(l))]\left[H_{c}\left(\operatorname{Ig}_{b}, \overline{\mathbb{Q}}_{l}\right)\right][Hc(Igb,Q¯l)] in the Grothendieck group of G ( A , p ) × J b ( Q p ) G A ∞ , p × J b Q p G(A^(oo,p))xxJ_(b)(Q_(p))G\left(\mathbb{A}^{\infty, p}\right) \times J_{b}\left(\mathbb{Q}_{p}\right)G(A∞,p)×Jb(Qp) representations. (The precise condition has to do with twisting by a high power of Frobenius in the Fujiwara-Varshavsky formula.) The proof of the theorem proceeds by carefully adapting the methods of [ 29 , 31 ] [ 29 , 31 ] [29,31][29,31][29,31] but with significant changes occurring at p p ppp, thus often requiring different techniques and arguments.
Formula (7.2) was obtained for some simple PEL-type Shimura varieties in [21, cHAP. 5] and [63] without formulating and proving the LR conjecture. In contrast, the above theorem represents the first LR-style approach to Igusa varieties, giving it two advantages. Firstly, the new approach makes the similarities between Shimura and Igusa varieties transparent. An important consequence is that the hard-won statement ( L R 0 ) L R 0 (LR_(0))\left(\mathrm{LR}_{0}\right)(LR0) for Shimura varieties can be transferred to the Igusa side. (If we had the full (LR) for Shimura varieties, then that would carry over to Igusa varieties, too.) Going from ( L R 0 ) L R 0 (LR_(0))\left(\mathrm{LR}_{0}\right)(LR0) for Igusa varieties to (7.2) is mostly the same as for Shimura varieties. Secondly, just like for Shimura varieties, the LR-style approach makes it feasible to extend the theorem to the abelian-type case, cf. Remark 3.4. (This extension has not been worked out, yet.) It should also be possible to go beyond good reduction and work in the setup of Kisin-Pappas models (Section 6). For the sake of proposing a problem, we can be even more general but still stick to ( L R 0 ) L R 0 (LR_(0))\left(\mathrm{LR}_{0}\right)(LR0) rather than (LR) as this should suffice for most applications:
Problem 7.3. Construct Igusa varieties modulo p p ppp for all Shimura data ( G , X ) ( G , X ) (G,X)(G, X)(G,X) and all primes p p ppp. Prove the ( L R 0 ) L R 0 (LR_(0))\left(\mathrm{LR}_{0}\right)(LR0)-version of Conjecture 7.1, thereby deduce formula (7.2).
Assuming a positive answer (known in the setting of Theorem 7.2), the next step is to unconditionally stabilize (7.2) into the following form:
(7.3) tr ( f , p × ϕ p [ H c ( I g b , Q ¯ l ) ] ) = e E e l l ( G ) S T e l l e ( f e , , p f p e , 1 f e ) (7.3) tr ⁡ f ∞ , p × Ï• p ′ ∣ H c I g b , Q ¯ l = ∑ e ∈ E e l l ( G )   S T e l l e f e , ∞ , p f p e , 1 f ∞ e {:(7.3)tr(f^(oo,p)xxphi_(p)^(')∣[H_(c)(Ig_(b), bar(Q)_(l))])=sum_(e inE_(ell)(G))ST_(ell)^(e)(f^(e,oo,p)f_(p)^(e,1)f_(oo)^(e)):}\begin{equation*} \operatorname{tr}\left(f^{\infty, p} \times \phi_{p}^{\prime} \mid\left[H_{c}\left(\mathrm{Ig}_{b}, \overline{\mathbb{Q}}_{l}\right)\right]\right)=\sum_{e \in \mathcal{E}_{\mathrm{ell}}(G)} \mathrm{ST}_{\mathrm{ell}}^{e}\left(f^{e, \infty, p} f_{p}^{e, 1} f_{\infty}^{e}\right) \tag{7.3} \end{equation*}(7.3)tr⁡(f∞,p×ϕp′∣[Hc(Igb,Q¯l)])=∑e∈Eell(G)STelle(fe,∞,pfpe,1f∞e)
The formula is an exact analogue of (3.8). Indeed, f e , , p f e , ∞ , p f^(e,oo,p)f^{e, \infty, p}fe,∞,p and f e f ∞ e f_(oo)^(e)f_{\infty}^{e}f∞e are constructed in the same way. However, f p e , f p e , f_(p)^(e,)f_{p}^{e,}fpe, is constructed from ϕ p Ï• p ′ phi_(p)^(')\phi_{p}^{\prime}Ï•p′ via a "nonstandard" transfer of functions this is the main novelty in the stabilization. Even when G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp is a product of general linear groups so that local endoscopy at p p ppp disappears, the transfer goes to G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp from an inner form of a Levi subgroup of G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp. The transfer was constructed and studied in [64] in a somewhat ad hoc manner, and later streamlined in [2]. Unfortunately, both papers make a set of technical hypotheses, to be removed in the work in progress with Bertoloni Meli.

7.2. Applications

The stabilization (7.3) is a significant step towards the following:
Problem 7.4. Obtain a decomposition of [ H c ( Ig b , Q ¯ l ) ] H c Ig b , Q ¯ l [H_(c)(Ig_(b), bar(Q)_(l))]\left[H_{c}\left(\operatorname{Ig}_{b}, \overline{\mathbb{Q}}_{l}\right)\right][Hc(Igb,Q¯l)] according to automorphic representations of G G GGG and its endoscopic groups, and describe each piece in the decomposition.
Just from the definition of Igusa varieties, it is not even clear whether the entirety of [ H c ( Ig b , Q ¯ l ) ] H c Ig b , Q ¯ l [H_(c)(Ig_(b), bar(Q)_(l))]\left[H_{c}\left(\operatorname{Ig}_{b}, \overline{\mathbb{Q}}_{l}\right)\right][Hc(Igb,Q¯l)] can be understood through automorphic representations. For Shimura varieties over C C C\mathbb{C}C, the connection is made through Matsushima's formula and its generalizations, but there is no analogue for Igusa varieties.
We have a concrete answer for some simple PEL-type Shimura varieties arising from ( G , X ) ( G , X ) (G,X)(G, X)(G,X) such that (i) endoscopy for G G GGG disappears over Q Q Q\mathbb{Q}Q and Q p Q p Q_(p)\mathbb{Q}_{p}Qp, (ii) G G GGG is anisotropic modulo center over Q Q Q\mathbb{Q}Q, and (iii) G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp is a product of general linear groups. Recall that J b J b J_(b)J_{b}Jb is an inner form of a Q p Q p Q_(p)\mathbb{Q}_{p}Qp-rational Levi subgroup, say M b M b M_(b)M_{b}Mb, of G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp. In fact, b b bbb determines a particular parabolic subgroup P b o p P b o p P_(b)^(op)P_{b}^{\mathrm{op}}Pbop containing M b M b M_(b)M_{b}Mb as a Levi component. Write Red b b ^(b){ }^{b}b for the composite morphism on the Grothendieck group of representations
Red b : Groth ( G ( Q p ) ) Groth ( M b ( Q p ) ) Groth ( J b ( Q p ) ) Red b : Groth ⁡ G Q p → Groth ⁡ M b Q p → Groth ⁡ J b Q p Red^(b):Groth(G(Q_(p)))rarr Groth(M_(b)(Q_(p)))rarr Groth(J_(b)(Q_(p)))\operatorname{Red}^{b}: \operatorname{Groth}\left(G\left(\mathbb{Q}_{p}\right)\right) \rightarrow \operatorname{Groth}\left(M_{b}\left(\mathbb{Q}_{p}\right)\right) \rightarrow \operatorname{Groth}\left(J_{b}\left(\mathbb{Q}_{p}\right)\right)Redb:Groth⁡(G(Qp))→Groth⁡(Mb(Qp))→Groth⁡(Jb(Qp))
where the first map is the Jacquet module relative to P b op P b op  P_(b)^("op ")P_{b}^{\text {op }}Pbop  (up to a character twist), and the second is Badulescu's Langlands-Jacquet map. The answer to Problem 7.4 in this setting is given by [21, тнм. V.5.4] and [66, тнм. 6.7]:
(7.5) [ H c ( Ig b , Q ¯ l ) ] = [ Red b H c ( Sh , Q ¯ l ) ] in Groth ( G ( A , p ) × J b ( Q p ) ) (7.5) H c Ig b , Q ¯ l = Red b ⁡ H c Sh , Q ¯ l  in  Groth ⁡ G A ∞ , p × J b Q p {:(7.5)[H_(c)(Ig_(b), bar(Q)_(l))]=[Red^(b)H_(c)(Sh, bar(Q)_(l))]quad" in "Groth(G(A^(oo,p))xxJ_(b)(Q_(p))):}\begin{equation*} \left[H_{c}\left(\operatorname{Ig}_{b}, \overline{\mathbb{Q}}_{l}\right)\right]=\left[\operatorname{Red}^{b} H_{c}\left(\operatorname{Sh}, \overline{\mathbb{Q}}_{l}\right)\right] \quad \text { in } \operatorname{Groth}\left(G\left(\mathbb{A}^{\infty, p}\right) \times J_{b}\left(\mathbb{Q}_{p}\right)\right) \tag{7.5} \end{equation*}(7.5)[Hc(Igb,Q¯l)]=[Redb⁡Hc(Sh,Q¯l)] in Groth⁡(G(A∞,p)×Jb(Qp))
Since H c ( S h , Q ¯ l ) H c S h , Q ¯ l H_(c)(Sh, bar(Q)_(l))H_{c}\left(\mathrm{Sh}, \overline{\mathbb{Q}}_{l}\right)Hc(Sh,Q¯l) is well understood by Matsushima's formula via relative Lie algebra cohomology, (7.5) is indeed a satisfactory answer for [ H c ( Ig b , Q ¯ l ) ] H c Ig b , Q ¯ l [H_(c)(Ig_(b), bar(Q)_(l))]\left[H_{c}\left(\operatorname{Ig}_{b}, \overline{\mathbb{Q}}_{l}\right)\right][Hc(Igb,Q¯l)]. Through Mantovan's formula, (7.5) sheds light on the cohomology of Rapoport-Zink spaces [3,66], cf. Section 5.
When G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp is still a product of general linear groups but G G GGG exhibits endoscopy over Q Q Q\mathbb{Q}Q, the formula for [ H c ( Ig b , Q ¯ l ) ] H c Ig b , Q ¯ l [H_(c)(Ig_(b), bar(Q)_(l))]\left[H_{c}\left(\operatorname{Ig}_{b}, \overline{\mathbb{Q}}_{l}\right)\right][Hc(Igb,Q¯l)] is no longer as simple as (7.5). Computing the formula in certain endoscopic cases was crucial in the proof of local-global compatibility in [65], cf. Section 5. (See [ 67 , $ 6 ] [ 67 , $ 6 ] [67,$6][67, \$ 6][67,$6] for an expository account.)
In general, Problem 7.4 seems out of reach. Firstly, just like for Shimura varieties, the lack of endoscopic classification is a major obstacle. Secondly, a new difficulty in the Igusa setup is that the analogue of Problem 4.1 has no conceptual reason to have a positive answer (cf. last paragraph of Section 4.1), and the analogue of Problem 4.2 is even less clear. (The second point is related to the question at the end of this section.) Assuming that both issues
go away, a conjectural formula for [ H c ( I g b , Q ¯ l ) ] H c I g b , Q ¯ l [H_(c)(Ig_(b), bar(Q)_(l))]\left[H_{c}\left(\mathrm{Ig}_{b}, \overline{\mathbb{Q}}_{l}\right)\right][Hc(Igb,Q¯l)] has been given in [2] under some hypotheses on G G GGG, resembling Kottwitz's conjectural formula for [ IH ( S h ¯ , Q ¯ l ) ] IH ⁡ S h ¯ , Q ¯ l [IH( bar(Sh), bar(Q)_(l))]\left[\operatorname{IH}\left(\overline{\mathrm{Sh}}, \overline{\mathbb{Q}}_{l}\right)\right][IH⁡(Sh¯,Q¯l)] in [32, §10]. The formula for [ H c ( I g b , Q ¯ l ) ] H c I g b , Q ¯ l [H_(c)(Ig_(b), bar(Q)_(l))]\left[H_{c}\left(\mathrm{Ig}_{b}, \overline{\mathbb{Q}}_{l}\right)\right][Hc(Igb,Q¯l)] is far more complicated than (7.5) and involves endoscopic versions of Red b Red b Red^(b)\operatorname{Red}^{b}Redb. (In the stable case, which is the simplest, the correct analogue of (7.4) is the Jacquet module followed by a stable transfer between inner forms.) A main observation in [2] is that the endoscopic versions of Red b Red b Red^(b)\operatorname{Red}^{b}Redb should interact with the cohomology of Rapoport-Zink spaces (and their generalizations) in an interesting way by a global reason.
For an unconditional result towards Problem 7.4, we managed to compute the G ( A , p ) × J b ( Q p ) G A ∞ , p × J b Q p G(A^(oo,p))xxJ_(b)(Q_(p))G\left(\mathbb{A}^{\infty, p}\right) \times J_{b}\left(\mathbb{Q}_{p}\right)G(A∞,p)×Jb(Qp)-module H 0 ( Ig b , Q ¯ l ) H 0 Ig b , Q ¯ l H^(0)(Ig_(b), bar(Q)_(l))H^{0}\left(\operatorname{Ig}_{b}, \overline{\mathbb{Q}}_{l}\right)H0(Igb,Q¯l) for unramified Shimura data ( G , X , p , E ) ( G , X , p , E ) (G,X,p,E)(G, X, p, \mathcal{E})(G,X,p,E) of Hodge type in joint work with Kret [36], when b b bbb is nonbasic 5 5 ^(5){ }^{5}5 (in every Q Q Q\mathbb{Q}Q-simple factor of the adjoint group of G G GGG ). In analogy with (7.5), the result may be expressed as
(7.6) H 0 ( Ig b , Q ¯ l ) = Red b H 0 ( Sh , Q ¯ l ) as G ( A , p ) × J b ( Q p ) -modules. (7.6) H 0 Ig b , Q ¯ l = Red b ⁡ H 0 Sh , Q ¯ l  as  G A ∞ , p × J b Q p -modules.  {:(7.6)H^(0)(Ig_(b), bar(Q)_(l))=Red^(b)H^(0)(Sh, bar(Q)_(l))quad" as "quad G(A^(oo,p))xxJ_(b)(Q_(p))"-modules. ":}\begin{equation*} H^{0}\left(\operatorname{Ig}_{b}, \overline{\mathbb{Q}}_{l}\right)=\operatorname{Red}^{b} H^{0}\left(\operatorname{Sh}, \overline{\mathbb{Q}}_{l}\right) \quad \text { as } \quad G\left(\mathbb{A}^{\infty, p}\right) \times J_{b}\left(\mathbb{Q}_{p}\right) \text {-modules. } \tag{7.6} \end{equation*}(7.6)H0(Igb,Q¯l)=Redb⁡H0(Sh,Q¯l) as G(A∞,p)×Jb(Qp)-modules. 
Here H 0 ( S h , Q ¯ l ) H 0 S h , Q ¯ l H^(0)(Sh, bar(Q)_(l))H^{0}\left(\mathrm{Sh}, \overline{\mathbb{Q}}_{l}\right)H0(Sh,Q¯l) has a well-known description in terms of 1-dimensional automorphic representations of G ( A ) G ( A ) G(A)G(\mathbb{A})G(A), and Red b Red b Red^(b)\operatorname{Red}^{b}Redb is unequivocally defined for 1-dimensional representations of G ( Q p ) G Q p G(Q_(p))G\left(\mathbb{Q}_{p}\right)G(Qp) (which are always stable). The proof starts from the results of Section 7.1. The main point is to get around the two essential difficulties mentioned in the last paragraph, by incorporating asymptotic and inductive arguments to extract the H 0 H 0 H^(0)H^{0}H0-part from a very complicated identity coming from (stabilized) trace formulas. The study of H 0 H 0 H^(0)H^{0}H0 was motivated by geometric applications to the irreducibility of Igusa varieties and to the discrete Hecke orbit conjecture. The reader is referred to [36] for details and further references. Similar geometric results were independently obtained by van Hoften and Xiao [ 68 , 69 [ 68 , 69 [68,69[68,69[68,69 ] via a more geometric approach without using automorphic forms or trace formulas.
We conclude this section with an unrefined question. Igusa varieties (at finite level) are almost never proper varieties, so the answer to Problem 7.4 does not determine H c i ( Ig b , Q ¯ l ) H c i Ig b , Q ¯ l H_(c)^(i)(Ig_(b), bar(Q)_(l))H_{c}^{i}\left(\operatorname{Ig}_{b}, \overline{\mathbb{Q}}_{l}\right)Hci(Igb,Q¯l) for i 0 i ≥ 0 i >= 0i \geq 0i≥0 due to possible cancelations in the Grothendieck group. (For improper Shimura varieties, the intersection cohomology is free from such a cancelation thanks to purity.) Thus we can ask whether there are useful compactifications of Igusa varieties to help us understand the cohomology more precisely. This was undertaken by Mantovan [45] in a special case (with a different goal). In general, it is unclear how to proceed even when Shimura varieties are proper. If a strategy is found in that case, it may be possible to deal with improperness of Shimura varieties by virture of Caraiani-Scholze's partial compactification [ 8 ] [ 8 ] [8][8][8].

ACKNOWLEDGMENTS

Special thanks are due to Michael Harris, Robert Kottwitz, and Richard Taylor for introducing the author to Shimura varieties and the Langlands program, and patiently answering his questions over many years. The author is grateful to Mark Kisin, Arno Kret, and Yihang Zhu for sharing their insight through years of recent and ongoing collaboration
5 If b b bbb is basic then Igusa varieties are 0 -dimensional and the picture is quite different from (7.6), cf. [36, §1.6]
that this paper is largely based on. He is also thankful for exciting mathematical journeys to the other coworkers: Sara Arias-de Reyna, Ana Caraiani, Luis Dieulefait, Matthew Emerton, Jessica Fintzen, Toby Gee, David Geraghty, Wushi Goldring, Julia Gordon, Junehyuk Jung, Tasho Kaletha, Julee Kim, Keerthi Madapusi Pera, Simon Marshall, Alberto Minguez, Vytautas Paškūnas, Peter Sarnak, Peter Scholze, Nicolas Templier, PaulJames White, and Gabor Wiese. Finally, he thanks Alex Youcis for comments to improve the exposition of the paper.

FUNDING

This work was partially supported by NSF grant DMS-1802039/2101688, NSF RTG grant DMS-1646385, and a Miller Professorship.

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SUG WOO SHIN

Department of Mathematics, UC Berkeley, Berkeley, CA 94720, USA, and Korea Institute for Advanced Study, Seoul 02455, Republic of Korea, sug.woo.shin@berkeley.edu

THE CONGRUENT NUMBER PROBLEM AND ELLIPTIC CURVES

YE TIAN

ABSTRACT

The Birch and Swinnerton-Dyer (BSD) conjecture and Goldfeld conjecture are fundamental problems in the arithmetic of elliptic curves. The congruent number problem (CNP) is one of the oldest problems in number theory which is, for each integer n n nnn, to find all the rational right triangles of area n n nnn. It is equivalent to finding all rational points on the elliptic curve E ( n ) : n y 2 = x 3 x E ( n ) : n y 2 = x 3 − x E^((n)):ny^(2)=x^(3)-xE^{(n)}: n y^{2}=x^{3}-xE(n):ny2=x3−x. The BSD conjecture for E ( n ) E ( n ) E^((n))E^{(n)}E(n) solves CNP, and Goldfeld conjecture for this elliptic curve family solves CNP for integers with probability one. In this article, we introduce some recent progress on these conjectures and problems.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 11-06; Secondary 11G05, 11G40

KEYWORDS

Congruent numbers, BSD conjecture, Goldfeld conjecture, quadratic twists, Selmer group, L-function

1. CONGRUENT NUMBER PROBLEM

A positive rational number n n nnn is called a congruent number if the following equivalent conditions hold:
(i) There exists a rational number x x xxx such that x 2 ± n x 2 ± n x^(2)+-nx^{2} \pm nx2±n are squares of rational numbers.
(ii) There exists a right triangle with rational side lengths (called a rational right triangle) whose area is n n nnn.
In his book Liber Quadratorum published in 1225, Fibonacci (1175-1250) named an integer satisfying (i) a "congruum" from the Latin, which means to meet together, since the three squares x 2 n , x 2 x 2 − n , x 2 x^(2)-n,x^(2)x^{2}-n, x^{2}x2−n,x2, and x 2 + n x 2 + n x^(2)+nx^{2}+nx2+n are congruent modulo n n nnn.
The congruent number problem (CNP, for short) is to determine, in finitely many steps, whether or not a given rational number is a congruent number, and, if it is, find all the corresponding x x xxx in (i) or rational right triangles in (ii). No such algorithm has ever been found. The Persian mathematician Al-Karaji (953-1029), perhaps the first mathematician, stated this problem in terms of (i). A similar question appeared in his Arabic translation of the work of Diophantus in Greek. In an Arab manuscript of the tenth century, Mohammed Ben Alhocain realized the equivalence between (i) and (ii) and stated that this problem is "the principal object of the theory of rational right triangles" (see Dickson's book [20, chAP.
XVI, P. 459]).
Recall that any rational Pythagorean triple has the following form:
2 a b t , ( a 2 b 2 ) t , ( a 2 + b 2 ) t 2 a b t , a 2 − b 2 t , a 2 + b 2 t 2abt,quad(a^(2)-b^(2))t,quad(a^(2)+b^(2))t2 a b t, \quad\left(a^{2}-b^{2}\right) t, \quad\left(a^{2}+b^{2}\right) t2abt,(a2−b2)t,(a2+b2)t
for a unique ( a , b , t ) ( a , b , t ) (a,b,t)(a, b, t)(a,b,t), where t t ttt is a positive rational number and a > b a > b a > ba>ba>b are two coprime positive integers with 2 ( a + b ) 2 ∤ ( a + b ) 2∤(a+b)2 \nmid(a+b)2∤(a+b). We call a rational Pythagorean triple primitive if t = 1 t = 1 t=1t=1t=1, i.e., its triangle has coprime integral side lengths. It follows that n n nnn is a congruent number if and only if n n nnn has the same square-free part as a b ( a + b ) ( a b ) a b ( a + b ) ( a − b ) ab(a+b)(a-b)a b(a+b)(a-b)ab(a+b)(a−b), for some integers a a aaa and b b bbb. For example, by taking ( a , b ) = ( 5 , 4 ) , ( 2 , 1 ) ( a , b ) = ( 5 , 4 ) , ( 2 , 1 ) (a,b)=(5,4),(2,1)(a, b)=(5,4),(2,1)(a,b)=(5,4),(2,1), and ( 16 , 9 ) ( 16 , 9 ) (16,9)(16,9)(16,9), note that 5 , 6 , 7 5 , 6 , 7 5,6,75,6,75,6,7 are congruent numbers with corresponding triangles ( 20 / 3 , 3 / 2 , 41 / 6 ) , ( 3 , 4 , 5 ) ( 20 / 3 , 3 / 2 , 41 / 6 ) , ( 3 , 4 , 5 ) (20//3,3//2,41//6),(3,4,5)(20 / 3,3 / 2,41 / 6),(3,4,5)(20/3,3/2,41/6),(3,4,5), and ( 24 / 5 , 35 / 12 , 337 / 60 ) ( 24 / 5 , 35 / 12 , 337 / 60 ) (24//5,35//12,337//60)(24 / 5,35 / 12,337 / 60)(24/5,35/12,337/60). To consider CNP, it is enough to consider square-free integers. In Liber Quadratorum, Fibonacci constructed these right triangles and also claimed that 1 is not a congruent number, but did not give a proof.
In 1640, Fermat discovered his infinite descent method to show that 1 , 2 , 3 1 , 2 , 3 1,2,31,2,31,2,3 are noncongruent numbers. The same method could be employed to find more noncongruent numbers, for example, any prime p 3 ( mod 8 ) p ≡ 3 ( mod 8 ) p-=3(mod8)p \equiv 3(\bmod 8)p≡3(mod8). In fact, suppose such a prime p p ppp is a congruent number, then there exists a primitive Pythagorean triple ( a 2 b 2 , 2 a b , a 2 + b 2 ) a 2 − b 2 , 2 a b , a 2 + b 2 (a^(2)-b^(2),2ab,a^(2)+b^(2))\left(a^{2}-b^{2}, 2 a b, a^{2}+b^{2}\right)(a2−b2,2ab,a2+b2) whose area a b ( a + b ) ( a b ) a b ( a + b ) ( a − b ) ab(a+b)(a-b)a b(a+b)(a-b)ab(a+b)(a−b) has the square-free part p p ppp. Assume the area is minimal. Since a , b , a + b , a b a , b , a + b , a − b a,b,a+b,a-ba, b, a+b, a-ba,b,a+b,a−b are coprime to each other, by modulo 8 consideration, we have
a = r 2 , b = p s 2 , a + b = u 2 , a b = v 2 a = r 2 , b = p s 2 , a + b = u 2 , a − b = v 2 a=r^(2),quad b=ps^(2),quad a+b=u^(2),quad a-b=v^(2)a=r^{2}, \quad b=p s^{2}, \quad a+b=u^{2}, \quad a-b=v^{2}a=r2,b=ps2,a+b=u2,a−b=v2
for some positive integers r , s , u , v r , s , u , v r,s,u,vr, s, u, vr,s,u,v. Note that the Pythagorean triple ( u v , u + v , 2 r ) ( u − v , u + v , 2 r ) (u-v,u+v,2r)(u-v, u+v, 2 r)(u−v,u+v,2r) is with smaller area, a contradiction.
More examples of congruent and noncongruent numbers (gray for non-congruent numbers) were found:
n mod 8 n mod 8 n mod8n \bmod 8nmod8 1 2 3 5 6 7
n n nnn 1 2 3 5 6 7
9 10 11 13 14 15
17 18 19 21 22 23
25 26 27 29 30 31
33 34 35 37 38 39
41 42 43 45 46 47
â‹® vdots\vdotsâ‹® â‹® vdots\vdotsâ‹® â‹® vdots\vdotsâ‹® â‹® vdots\vdotsâ‹® â‹® vdots\vdotsâ‹® â‹® vdots\vdotsâ‹®
217 218 219 221 222 223
â‹® vdots\vdotsâ‹® â‹® vdots\vdotsâ‹® â‹® vdots\vdotsâ‹® â‹® vdots\vdotsâ‹® â‹® vdots\vdotsâ‹® â‹® vdots\vdotsâ‹®
n mod8 1 2 3 5 6 7 n 1 2 3 5 6 7 9 10 11 13 14 15 17 18 19 21 22 23 25 26 27 29 30 31 33 34 35 37 38 39 41 42 43 45 46 47 vdots vdots vdots vdots vdots vdots 217 218 219 221 222 223 vdots vdots vdots vdots vdots vdots| $n \bmod 8$ | 1 | 2 | 3 | 5 | 6 | 7 | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | $n$ | 1 | 2 | 3 | 5 | 6 | 7 | | | 9 | 10 | 11 | 13 | 14 | 15 | | | 17 | 18 | 19 | 21 | 22 | 23 | | | 25 | 26 | 27 | 29 | 30 | 31 | | | 33 | 34 | 35 | 37 | 38 | 39 | | | 41 | 42 | 43 | 45 | 46 | 47 | | | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | | | 217 | 218 | 219 | 221 | 222 | 223 | | | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ |
From the table, one may conjecture that all the positive integers congruent to 5 , 6 , 7 5 , 6 , 7 5,6,75,6,75,6,7 modulo 8 are congruent numbers (conjectured by Alter, Curtz, and Kubota in [2]) and the density of positive integers congruent to 1 , 2 , 3 1 , 2 , 3 1,2,31,2,31,2,3 modulo 8 being non-congruent is one.
The arithmetic of elliptic curves, in particular the BSD conjecture and the Goldfeld conjecture, provides a systematical and deeper point of view to study CNP. We now recall these conjectures to introduce notation.
For an elliptic curve A A AAA over a number field F F FFF, the set A ( F ) A ( F ) A(F)A(F)A(F) of rational points has a finitely generated abelian group structure by Mordell-Weil theorem. Its rank is denoted by rank Z A ( F ) rank Z ⁡ A ( F ) rank_(Z)A(F)\operatorname{rank}_{\mathbb{Z}} A(F)rankZ⁡A(F). The Hasse-Weil L-function L ( s , A / F ) L ( s , A / F ) L(s,A//F)L(s, A / F)L(s,A/F) of A A AAA is defined as an Euler product and conjectured to be entire and to satisfy a functional equation. The vanishing order of L ( s , A / F ) L ( s , A / F ) L(s,A//F)L(s, A / F)L(s,A/F) at s = 1 s = 1 s=1s=1s=1 denoted by ord s = 1 L ( s , A / F ) ord s = 1 ⁡ L ( s , A / F ) ord_(s=1)L(s,A//F)\operatorname{ord}_{s=1} L(s, A / F)ords=1⁡L(s,A/F) is called the analytic rank of A / F A / F A//FA / FA/F. When F = Q F = Q F=QF=\mathbb{Q}F=Q, the conjecture is known by the work of Wiles [54], et al., and the functional equation is given by
Λ ( s , A / Q ) := N A s / 2 2 ( 2 π ) s Γ ( s ) L ( s , A / Q ) = ϵ ( A ) Λ ( 2 s , A / Q ) Λ ( s , A / Q ) := N A s / 2 â‹… 2 ( 2 Ï€ ) − s Γ ( s ) L ( s , A / Q ) = ϵ ( A ) Λ ( 2 − s , A / Q ) Lambda(s,A//Q):=N_(A)^(s//2)*2(2pi)^(-s)Gamma(s)L(s,A//Q)=epsilon(A)Lambda(2-s,A//Q)\Lambda(s, A / \mathbb{Q}):=N_{A}^{s / 2} \cdot 2(2 \pi)^{-s} \Gamma(s) L(s, A / \mathbb{Q})=\epsilon(A) \Lambda(2-s, A / \mathbb{Q})Λ(s,A/Q):=NAs/2â‹…2(2Ï€)−sΓ(s)L(s,A/Q)=ϵ(A)Λ(2−s,A/Q)
where N A Z 1 N A ∈ Z ≥ 1 N_(A)inZ_( >= 1)N_{A} \in \mathbb{Z}_{\geq 1}NA∈Z≥1 is the conductor of A / Q A / Q A//QA / \mathbb{Q}A/Q, and ϵ ( A ) { ± 1 } ϵ ( A ) ∈ { ± 1 } epsilon(A)in{+-1}\epsilon(A) \in\{ \pm 1\}ϵ(A)∈{±1} is the root number.
Conjecture 1 (BSD). Let A be an elliptic curve over a number field F F FFF. Then the following holds:
(1) rank Z A ( F ) = ord s = 1 L ( s , A / F ) rank Z ⁡ A ( F ) = ord s = 1 ⁡ L ( s , A / F ) rank_(Z)A(F)=ord_(s=1)L(s,A//F)\operatorname{rank}_{\mathbb{Z}} A(F)=\operatorname{ord}_{s=1} L(s, A / F)rankZ⁡A(F)=ords=1⁡L(s,A/F).
(2) The Tate-Shafarevich group ⨿ ( A / F ) ⨿ ( A / F ) ⨿(A//F)\amalg(A / F)⨿(A/F) is finite. For r = ord s = 1 L ( s , A / F ) r = ord s = 1 ⁡ L ( s , A / F ) r=ord_(s=1)L(s,A//F)r=\operatorname{ord}_{s=1} L(s, A / F)r=ords=1⁡L(s,A/F),
L ( r ) ( 1 , A / F ) r ! Ω A R A / | D F | = v c v # ⨿ ( A / F ) # A ( F ) t o r s 2 L ( r ) ( 1 , A / F ) r ! Ω A R A / D F = ∏ v   c v â‹… # ⨿ ( A / F ) # A ( F ) t o r s 2 (L^((r))(1,A//F))/(r!Omega_(A)R_(A)//sqrt(|D_(F)|))=(prod_(v)c_(v)*#⨿(A//F))/(#A(F)_(tors)^(2))\frac{L^{(r)}(1, A / F)}{r!\Omega_{A} R_{A} / \sqrt{\left|D_{F}\right|}}=\frac{\prod_{v} c_{v} \cdot \# \amalg(A / F)}{\# A(F)_{\mathrm{tors}}^{2}}L(r)(1,A/F)r!ΩARA/|DF|=∏vcvâ‹…#⨿(A/F)#A(F)tors2
Here D F D F D_(F)D_{F}DF is the discriminant of F F FFF, while Ω A , R A Ω A , R A Omega_(A),R_(A)\Omega_{A}, R_{A}ΩA,RA, and c v c v c_(v)c_{v}cv are the Néron period, the regulator, and the Tamagawa number of A A AAA at place v v vvv, respectively.
For a prime p p ppp, we call the equality of p p ppp-valuation on both sides the p p ppp-part BSD formula.
One significant fact related to the BSD conjecture for an elliptic curve A A AAA over Q Q Q\mathbb{Q}Q is that if it holds, then there will be an effective algorithm to compute generators of A ( Q ) A ( Q ) A(Q)A(\mathbb{Q})A(Q) [39]. It is easy to see that a positive integer n n nnn is a congruent number if and only if the elliptic curve (called a congruent elliptic curve)
E ( n ) : n y 2 = x 3 x E ( n ) : n y 2 = x 3 − x E^((n)):ny^(2)=x^(3)-xE^{(n)}: n y^{2}=x^{3}-xE(n):ny2=x3−x
has Mordell-Weil group E ( n ) ( Q ) E ( n ) ( Q ) E^((n))(Q)E^{(n)}(\mathbb{Q})E(n)(Q) of positive rank. There exists a one-to-one correspondence between rational right triangles with area n n nnn and nontorsion rational points of E ( n ) E ( n ) E^((n))E^{(n)}E(n). In particular, the BSD conjecture for E ( n ) E ( n ) E^((n))E^{(n)}E(n) would solve the CNP.
A fundamental result on the BSD conjecture was obtained by Coates-Wiles [18], Rubin [43], Gross-Zagier [24], and Kolyvagin [36]: If ord s = 1 L ( s , A / Q ) 1 s = 1 L ( s , A / Q ) ≤ 1 _(s=1)L(s,A//Q) <= 1{ }_{s=1} L(s, A / \mathbb{Q}) \leq 1s=1L(s,A/Q)≤1, then
rank Z A ( Q ) = ord s = 1 L ( s , A / Q ) rank Z ⁡ A ( Q ) = ord s = 1 ⁡ L ( s , A / Q ) rank_(Z)A(Q)=ord_(s=1)L(s,A//Q)\operatorname{rank}_{\mathbb{Z}} A(\mathbb{Q})=\operatorname{ord}_{s=1} L(s, A / \mathbb{Q})rankZ⁡A(Q)=ords=1⁡L(s,A/Q)
and # ⨿ ( A / Q ) < # ⨿ ( A / Q ) < ∞ #⨿(A//Q) < oo\# \amalg(A / \mathbb{Q})<\infty#⨿(A/Q)<∞. There are several results on the p p ppp-part BSD formula, including Rubin [43], Kato [33], Kolyvagin [36], Skinner-Urban [46], Zhang [56], Jetchev-Skinner-Wan [31]. The full BSD conjecture was verified for a subfamily of congruent elliptic curves, which have both algebraic and analytic rank one.
Theorem 2 ( [ 37 ] ) 2 ( [ 37 ] ) 2([37])2([37])2([37]). Let n 5 ( mod 8 ) n ≡ 5 ( mod 8 ) n-=5(mod8)n \equiv 5(\bmod 8)n≡5(mod8) be a square-free positive integer, all of whose prime factors are congruent to 1 modulo 4 . Assume that Q ( n ) Q ( − n ) Q(sqrt(-n))\mathbb{Q}(\sqrt{-n})Q(−n) has no ideal class of order 4 , then E ( n ) : y 2 = x 3 n 2 x E ( n ) : y 2 = x 3 − n 2 x E^((n)):y^(2)=x^(3)-n^(2)xE^{(n)}: y^{2}=x^{3}-n^{2} xE(n):y2=x3−n2x has both algebraic and analytic rank 1 and the full BSD conjecture holds.
For the above congruent number elliptic curves, the 2-part of the BSD formula is proved in [51], [50]. The p p ppp-part of the BSD formula, when p 3 , p n p ≥ 3 , p ∤ n p >= 3,p∤np \geq 3, p \nmid np≥3,p∤n, is the consequence of works by Perrin-Riou [42], Kobayashi [35], etal. The p p ppp-part of the BSD formula, when p 1 ( mod 4 ) , p n p ≡ 1 ( mod 4 ) , p ∣ n p-=1(mod4),p∣np \equiv 1(\bmod 4), p \mid np≡1(mod4),p∣n, is proved in Li-Liu-Tian [37]. The generalization of Kobayashi's work to potential supersingular primes together with the argument of Perrin-Riou [42], also implies the p p ppp-part BSD formula for primes p p ppp of potential supersingular reduction (see [41]).
There is a conjecture on statistical behaviors of analytic ranks for a quadratic twist family of elliptic curves. For an elliptic curve y 2 = f ( x ) y 2 = f ( x ) y^(2)=f(x)y^{2}=f(x)y2=f(x) over F F FFF, its quadratic twist family consists of elliptic curves n y 2 = f ( x ) n y 2 = f ( x ) ny^(2)=f(x)n y^{2}=f(x)ny2=f(x) with n F × n ∈ F × n inF^(xx)n \in F^{\times}n∈F×. Based on minimalist principle, Goldfeld proposed the following:
Conjecture 3 (Goldfeld [14,23]). Let ε { ± 1 } ε ∈ { ± 1 } epsi in{+-1}\varepsilon \in\{ \pm 1\}ε∈{±1} and A A A\mathscr{A}A be a quadratic twist family of elliptic curves over F F FFF. Then, ordered by norms of conductors, among the quadratic twists A A A ∈ A A inAA \in \mathcal{A}A∈A with ϵ ( A ) = ε ϵ ( A ) = ε epsilon(A)=epsi\epsilon(A)=\varepsilonϵ(A)=ε,
Prob ( ord s = 1 L ( s , A / F ) = 0 ) ( resp . Prob ( ord s = 1 L ( s , A / F ) = 1 ) ) Prob ⁡ ord s = 1 ⁡ L ( s , A / F ) = 0 resp . Prob ⁡ ord s = 1 ⁡ L ( s , A / F ) = 1 Prob(ord_(s=1)L(s,A//F)=0)quad(resp.Prob(ord_(s=1)L(s,A//F)=1))\operatorname{Prob}\left(\operatorname{ord}_{s=1} L(s, A / F)=0\right) \quad\left(\operatorname{resp} . \operatorname{Prob}\left(\operatorname{ord}_{s=1} L(s, A / F)=1\right)\right)Prob⁡(ords=1⁡L(s,A/F)=0)(resp.Prob⁡(ords=1⁡L(s,A/F)=1))
is one if ε = + 1 ε = + 1 epsi=+1\varepsilon=+1ε=+1 (resp. -1 ). In particular, if F = Q F = Q F=QF=\mathbb{Q}F=Q, as A runs over a quadratic twist family of elliptic curves,
Prob ( ord s = 1 L ( s , A / Q ) = r ) Prob ⁡ ord s = 1 ⁡ L ( s , A / Q ) = r Prob(ord_(s=1)L(s,A//Q)=r)\operatorname{Prob}\left(\operatorname{ord}_{s=1} L(s, A / \mathbb{Q})=r\right)Prob⁡(ords=1⁡L(s,A/Q)=r)
is equal to 1 / 2 1 / 2 1//21 / 21/2 for r = 0 , 1 r = 0 , 1 r=0,1r=0,1r=0,1, and 0 for r 2 r ≥ 2 r >= 2r \geq 2r≥2.
We refer to ϵ = 1 ϵ = 1 epsilon=1\epsilon=1ϵ=1 (resp. -1 ) case of the conjecture as the even (resp. odd) parity Goldfeld conjecture. The significance of Goldfeld conjecture is that, together with the GrossZagier formula (see Section 2), it solves the problem of finding generators of A ( Q ) A ( Q ) A(Q)A(\mathbb{Q})A(Q) for density-one elliptic curves A A AAA in a quadratic twist family.
Conjecture 4 (Goldfeld [23], Katz-Sarnak [34], etc.). Let A run over all elliptic curves over a fixed number field F F FFF as ordered by height, then
Prob ( ord s = 1 L ( s , A / F ) = r ) Prob ⁡ ord s = 1 ⁡ L ( s , A / F ) = r Prob(ord_(s=1)L(s,A//F)=r)\operatorname{Prob}\left(\operatorname{ord}_{s=1} L(s, A / F)=r\right)Prob⁡(ords=1⁡L(s,A/F)=r)
is equal to 1 / 2 1 / 2 1//21 / 21/2 for r = 0 , 1 r = 0 , 1 r=0,1r=0,1r=0,1, and 0 for r 2 r ≥ 2 r >= 2r \geq 2r≥2.
For n Z 0 n ∈ Z ≥ 0 n inZ_( >= 0)n \in \mathbb{Z}_{\geq 0}n∈Z≥0, we have
ϵ ( E ( n ) ) = { + 1 , n 1 , 2 , 3 ( mod 8 ) 1 , n 5 , 6 , 7 ( mod 8 ) ϵ E ( n ) = + 1 , n ≡ 1 , 2 , 3 ( mod 8 ) − 1 , n ≡ 5 , 6 , 7 ( mod 8 ) epsilon(E^((n)))={[+1",",n-=1","2","3(mod8)],[-1",",n-=5","6","7(mod8)]:}\epsilon\left(E^{(n)}\right)= \begin{cases}+1, & n \equiv 1,2,3(\bmod 8) \\ -1, & n \equiv 5,6,7(\bmod 8)\end{cases}ϵ(E(n))={+1,n≡1,2,3(mod8)−1,n≡5,6,7(mod8)
The central L L L\mathrm{L}L-value of E ( n ) E ( n ) E^((n))E^{(n)}E(n) is related to the following ternary quadratic equation by Tunnell [52]: For a positive square-free integer n n nnn, let a = 1 a = 1 a=1a=1a=1 if n n nnn is odd and a = 2 a = 2 a=2a=2a=2 if n n nnn is even. Consider the equation
2 a x 2 + y 2 + 8 z 2 = n / a , x , y , z Z 2 a x 2 + y 2 + 8 z 2 = n / a , x , y , z ∈ Z 2ax^(2)+y^(2)+8z^(2)=n//a,quad x,y,z inZ2 a x^{2}+y^{2}+8 z^{2}=n / a, \quad x, y, z \in \mathbb{Z}2ax2+y2+8z2=n/a,x,y,z∈Z
Let Σ ( n ) Σ ( n ) Sigma(n)\Sigma(n)Σ(n) be the set of its solutions and let
L ( n ) = # { ( x , y , z ) Σ ( n ) | 2 | z } # { ( x , y , z ) Σ ( n ) 2 z } L ( n ) = # { ( x , y , z ) ∈ Σ ( n ) | 2 | z } − # { ( x , y , z ) ∈ Σ ( n ) ∣ 2 ∤ z } L(n)=#{(x,y,z)in Sigma(n)|2|z}-#{(x,y,z)in Sigma(n)∣2∤z}\mathscr{L}(n)=\#\{(x, y, z) \in \Sigma(n)|2| z\}-\#\{(x, y, z) \in \Sigma(n) \mid 2 \nmid z\}L(n)=#{(x,y,z)∈Σ(n)|2|z}−#{(x,y,z)∈Σ(n)∣2∤z}
It is easy to see that L ( n ) = 0 L ( n ) = 0 L(n)=0\mathscr{L}(n)=0L(n)=0 for positive n 5 , 6 , 7 ( mod 8 ) n ≡ 5 , 6 , 7 ( mod 8 ) n-=5,6,7(mod8)n \equiv 5,6,7(\bmod 8)n≡5,6,7(mod8). Tunnell proved that for n n nnn positive square-free, L ( n ) 0 L ( n ) ≠ 0 L(n)!=0\mathscr{L}(n) \neq 0L(n)≠0 if and only if L ( 1 , E ( n ) ) 0 L 1 , E ( n ) ≠ 0 L(1,E^((n)))!=0L\left(1, E^{(n)}\right) \neq 0L(1,E(n))≠0. The BSD conjecture predicts the following:
Conjecture A. A positive square-free integer n n nnn is a congruent number if and only if L ( n ) = 0 L ( n ) = 0 L(n)=0\mathscr{L}(n)=0L(n)=0. In particular, any positive integer n 5 , 6 , 7 ( mod 8 ) n ≡ 5 , 6 , 7 ( mod 8 ) n-=5,6,7(mod8)n \equiv 5,6,7(\bmod 8)n≡5,6,7(mod8) is a congruent number.
One can determine whether L ( n ) = 0 L ( n ) = 0 L(n)=0\mathscr{L}(n)=0L(n)=0 in finitely many steps, yet there is no algorithm to find all the rational points of E ( n ) E ( n ) E^((n))E^{(n)}E(n). Tunnell's work was recently generalized to any given quadratic twist family of elliptic curves over Q Q Q\mathbb{Q}Q in [26].
The even Goldfeld conjecture for the family E ( n ) E ( n ) E^((n))E^{(n)}E(n) can be stated as follows:
Conjecture B1. Among all square-free positive integers n 1 , 2 , 3 ( mod 8 ) n ≡ 1 , 2 , 3 ( mod 8 ) n-=1,2,3(mod8)n \equiv 1,2,3(\bmod 8)n≡1,2,3(mod8), the subset of n n nnn with L ( n ) 0 L ( n ) ≠ 0 L(n)!=0\mathscr{L}(n) \neq 0L(n)≠0 has density one.
For an elliptic curve A / Q A / Q A//QA / \mathbb{Q}A/Q with root number -1 , the BSD conjecture predicts that A ( Q ) A ( Q ) A(Q)A(\mathbb{Q})A(Q) has an infinite-order point. Heegner point construction provides a systematic method to construct rational points. We now give a concrete construction for congruent elliptic curves E ( n ) E ( n ) E^((n))E^{(n)}E(n) with n 5 , 6 , 7 ( mod 8 ) n ≡ 5 , 6 , 7 ( mod 8 ) n-=5,6,7(mod8)n \equiv 5,6,7(\bmod 8)n≡5,6,7(mod8). Denote by E E EEE the elliptic curve y 2 = x 3 x y 2 = x 3 − x y^(2)=x^(3)-xy^{2}=x^{3}-xy2=x3−x that has conductor 32. The Abel-Jacobi map induces the complex uniformization
E ( C ) C / Λ E , z O z d x / 2 y E ( C ) ≃ C / Λ E , z ↦ ∫ O z   d x / 2 y E(C)≃C//Lambda_(E),quad z|->int_(O)^(z)dx//2yE(\mathbb{C}) \simeq \mathbb{C} / \Lambda_{E}, \quad z \mapsto \int_{O}^{z} d x / 2 yE(C)≃C/ΛE,z↦∫Ozdx/2y
where Λ E = { γ d x / 2 y γ H 1 ( E ( C ) , Z ) } C Λ E = ∫ γ   d x / 2 y ∣ γ ∈ H 1 ( E ( C ) , Z ) ⊂ C Lambda_(E)={int_(gamma)dx//2y∣gamma inH_(1)(E(C),Z)}subC\Lambda_{E}=\left\{\int_{\gamma} d x / 2 y \mid \gamma \in H_{1}(E(\mathbb{C}), \mathbb{Z})\right\} \subset \mathbb{C}ΛE={∫γdx/2y∣γ∈H1(E(C),Z)}⊂C is the period lattice. Denote by ϕ Ï• phi\phiÏ• the newform of weight 2 and level Γ 0 ( 32 ) Γ 0 ( 32 ) Gamma_(0)(32)\Gamma_{0}(32)Γ0(32) associated to E E EEE. Let f f fff be the analytic map
f : X 0 ( 32 ) ( C ) E ( C ) f : X 0 ( 32 ) ( C ) → E ( C ) f:X_(0)(32)(C)rarr E(C)f: X_{0}(32)(\mathbb{C}) \rightarrow E(\mathbb{C})f:X0(32)(C)→E(C)
induced by the above complex uniformization E ( C ) C / Λ E E ( C ) ≃ C / Λ E E(C)≃C//Lambda_(E)E(\mathbb{C}) \simeq \mathbb{C} / \Lambda_{E}E(C)≃C/ΛE and
H C : τ i τ 2 π i ϕ ( z ) d z H → C : Ï„ ↦ ∫ i ∞ Ï„   2 Ï€ i Ï• ( z ) d z HrarrC:tau|->int_(i oo)^(tau)2pi i phi(z)dz\mathscr{H} \rightarrow \mathbb{C}: \tau \mapsto \int_{i \infty}^{\tau} 2 \pi i \phi(z) d zH→C:τ↦∫i∞τ2Ï€iÏ•(z)dz
We now give a construction of Heegner points. For n n nnn a positive square-free integer 5 , 6 , 7 ( mod 8 ) ≡ 5 , 6 , 7 ( mod 8 ) -=5,6,7(mod8)\equiv 5,6,7(\bmod 8)≡5,6,7(mod8), let K = Q ( n ) K = Q ( − n ) K=Q(sqrt(-n))K=\mathbb{Q}(\sqrt{-n})K=Q(−n), let O O O\mathcal{O}O be its ring of integers, and H H HHH its Hilbert class field. Let c c ccc be the complex conjugation and let E ( K ) c = 1 E ( K ) E ( K ) c = − 1 ⊂ E ( K ) E(K)^(c=-1)sub E(K)E(K)^{c=-1} \subset E(K)E(K)c=−1⊂E(K) be the subgroup on which c c ccc acts by -1 , then we naturally have E ( K ) c = 1 E ( n ) ( Q ) E ( K ) c = − 1 ≃ E ( n ) ( Q ) E(K)^(c=-1)≃E^((n))(Q)E(K)^{c=-1} \simeq E^{(n)}(\mathbb{Q})E(K)c=−1≃E(n)(Q).
Define the Heegner point, which lies in E ( n ) ( Q ) Q E ( n ) ( Q ) ⊗ Q E^((n))(Q)oxQE^{(n)}(\mathbb{Q}) \otimes \mathbb{Q}E(n)(Q)⊗Q, as follows:
y n := { tr H / K ( 1 [ i ] ) f ( τ n ) , tr H / K [ i ] f ( τ n ) , 2 tr H / K f ( τ n ) , where H τ n = { 1 4 ( 1 n ) , 1 4 n , 2 ϵ + n , if n { 5 ( mod 8 ) , 6 ( mod 8 ) , 7 ( mod 8 ) y n := tr H / K ⁡ ( 1 − [ i ] ) â‹… f Ï„ n , tr H / K ⁡ [ i ] â‹… f Ï„ n , 2 tr H / K ⁡ f Ï„ n ,  where  H ∋ Ï„ n = 1 4 ( 1 − − n ) , − 1 4 − n , − 2 ϵ + − n ,  if  n ≡ 5 ( mod 8 ) , 6 ( mod 8 ) , 7 ( mod 8 ) y_(n):={[tr_(H//K)(1-[i])*f(tau_(n))","],[tr_(H//K)[i]*f(tau_(n))","],[2tr_(H//K)f(tau_(n))","]quad" where "H∋tau_(n)={[(1)/(4(1-sqrt(-n)))","],[(-1)/(4sqrt(-n))","],[(-2)/(epsilon+sqrt(-n))","]quad" if "n-={[5(mod8)","],[6(mod8)","],[7(mod8)]:}y_{n}:=\left\{\begin{array}{l} \operatorname{tr}_{H / K}(1-[i]) \cdot f\left(\tau_{n}\right), \\ \operatorname{tr}_{H / K}[i] \cdot f\left(\tau_{n}\right), \\ 2 \operatorname{tr}_{H / K} f\left(\tau_{n}\right), \end{array} \quad \text { where } \mathscr{H} \ni \tau_{n}=\left\{\begin{array} { l } { \frac { 1 } { 4 ( 1 - \sqrt { - n } ) } , } \\ { \frac { - 1 } { 4 \sqrt { - n } } , } \\ { \frac { - 2 } { \epsilon + \sqrt { - n } } , } \end{array} \quad \text { if } n \equiv \left\{\begin{array}{l} 5(\bmod 8), \\ 6(\bmod 8), \\ 7(\bmod 8) \end{array}\right.\right.\right.yn:={trH/K⁡(1−[i])â‹…f(Ï„n),trH/K⁡[i]â‹…f(Ï„n),2trH/K⁡f(Ï„n), where H∋τn={14(1−−n),−14−n,−2ϵ+−n, if n≡{5(mod8),6(mod8),7(mod8)
Here ϵ ϵ epsilon\epsilonϵ is an integer such that ϵ 2 n ( mod 128 ) ϵ 2 ≡ − n ( mod 128 ) epsilon^(2)-=-n(mod 128)\epsilon^{2} \equiv-n(\bmod 128)ϵ2≡−n(mod128). The construction is natural from an automorphic representation point of view, which will be described in Section 2.
Conjecture B2. Among all positive integers n 5 , 6 , 7 ( mod 8 ) n ≡ 5 , 6 , 7 ( mod 8 ) n-=5,6,7(mod8)n \equiv 5,6,7(\bmod 8)n≡5,6,7(mod8), the subset of n n nnn with y n y n y_(n)y_{n}yn being nontorsion has density one.
The Gross-Zagier formula (see Section 2) implies that y n y n y_(n)y_{n}yn is nontorsion if and only if L ( 1 , E ( n ) ) 0 L ′ 1 , E ( n ) ≠ 0 L^(')(1,E^((n)))!=0L^{\prime}\left(1, E^{(n)}\right) \neq 0L′(1,E(n))≠0. Furthermore, Kolyvagin's work shows that if y n y n y_(n)y_{n}yn is nontorsion, then the rank of E ( n ) ( Q ) E ( n ) ( Q ) E^((n))(Q)E^{(n)}(\mathbb{Q})E(n)(Q) is one [36]. The Gross-Zagier formula also helps compute y n y n y_(n)y_{n}yn and therefore a generator of E ( n ) ( Q ) [ 19 ] E ( n ) ( Q ) [ 19 ] E^((n))(Q)[19]E^{(n)}(\mathbb{Q})[19]E(n)(Q)[19].
Remark 5. The combination of Conjectures B1 and B 2 B 2 B2B 2B2 is equivalent to Goldfeld conjecture for congruent elliptic curves, which would solve the CNP for integers with probability one.
Example 1. For n = 101 , 102 n = 101 , 102 n=101,102n=101,102n=101,102, and 103, the Heegner point y n y n y_(n)y_{n}yn is given by
( 3975302500 442723681 , 2808122994457950 9315348971921 ) , ( 5100 , 364140 ) − 3975302500 442723681 , 2808122994457950 9315348971921 , ( 5100 , 364140 ) ((-3975302500)/(442723681),(2808122994457950)/(9315348971921)),(5100,364140)\left(\frac{-3975302500}{442723681}, \frac{2808122994457950}{9315348971921}\right),(5100,364140)(−3975302500442723681,28081229944579509315348971921),(5100,364140)
and
( 777848715219380607 8780605285453456 , 406939902409963977921570495 822785599723202981879104 ) − 777848715219380607 8780605285453456 , 406939902409963977921570495 822785599723202981879104 ((-777848715219380607)/(8780605285453456),(406939902409963977921570495)/(822785599723202981879104))\left(\frac{-777848715219380607}{8780605285453456}, \frac{406939902409963977921570495}{822785599723202981879104}\right)(−7778487152193806078780605285453456,406939902409963977921570495822785599723202981879104)
And right triangles with area n n nnn corresponding to y n y n y_(n)y_{n}yn have side lengths
( 267980280100 44538033219 , 44538033219 1326635050 , 2015242462949760001961 59085715926389725950 ) , ( 20 7 , 357 5 , 2501 35 ) 267980280100 44538033219 , 44538033219 1326635050 , 2015242462949760001961 59085715926389725950 , 20 7 , 357 5 , 2501 35 ((267980280100)/(44538033219),(44538033219)/(1326635050),(2015242462949760001961)/(59085715926389725950)),((20)/(7),(357)/(5),(2501)/(35))\left(\frac{267980280100}{44538033219}, \frac{44538033219}{1326635050}, \frac{2015242462949760001961}{59085715926389725950}\right),\left(\frac{20}{7}, \frac{357}{5}, \frac{2501}{35}\right)(26798028010044538033219,445380332191326635050,201524246294976000196159085715926389725950),(207,3575,250135)
and
( 16286253110943816 441394452081515 , 45463628564396045 8143126555471908 , 134130664938047228374702001079697 3594330884182957394223708580620 ) 16286253110943816 441394452081515 , 45463628564396045 8143126555471908 , 134130664938047228374702001079697 3594330884182957394223708580620 ((16286253110943816)/(441394452081515),(45463628564396045)/(8143126555471908),(134130664938047228374702001079697)/(3594330884182957394223708580620))\left(\frac{16286253110943816}{441394452081515}, \frac{45463628564396045}{8143126555471908}, \frac{134130664938047228374702001079697}{3594330884182957394223708580620}\right)(16286253110943816441394452081515,454636285643960458143126555471908,1341306649380472283747020010796973594330884182957394223708580620)
Heegner [28] in 1952 showed that any prime or double prime n 5 , 6 , 7 ( mod 8 ) n ≡ 5 , 6 , 7 ( mod 8 ) n-=5,6,7(mod8)n \equiv 5,6,7(\bmod 8)n≡5,6,7(mod8) is a congruent number. Later on, based on Heegner's method, Monsky [40] in 1990 proved that for ( p 1 , p 2 ) ( 1 , 5 ) ( mod 8 ) ( resp . ( p 1 , p 2 ) ( 1 , 7 ) ( mod 8 ) ) p 1 , p 2 ≡ ( 1 , 5 ) ( mod 8 ) resp . p 1 , p 2 ≡ ( 1 , 7 ) ( mod 8 ) (p_(1),p_(2))-=(1,5)(mod8)(resp.(p_(1),p_(2))-=(1,7)(mod8))\left(p_{1}, p_{2}\right) \equiv(1,5)(\bmod 8)\left(\operatorname{resp} .\left(p_{1}, p_{2}\right) \equiv(1,7)(\bmod 8)\right)(p1,p2)≡(1,5)(mod8)(resp.(p1,p2)≡(1,7)(mod8)), two primes such that ( p 1 p 2 ) = 1 p 1 p 2 = − 1 ((p_(1))/(p_(2)))=-1\left(\frac{p_{1}}{p_{2}}\right)=-1(p1p2)=−1, the product p 1 p 2 p 1 p 2 p_(1)p_(2)p_{1} p_{2}p1p2 (resp. 2 p 1 p 2 2 p 1 p 2 2p_(1)p_(2)2 p_{1} p_{2}2p1p2 ) is a congruent number. A natural question is to seek congruent numbers with many prime factors. The following was first conjectured by Monsky in [40].
Theorem 6 (Tian [50]). Let n n nnn be the product of an odd number of primes 5 ( mod 8 ) ≡ 5 ( mod 8 ) -=5(mod8)\equiv 5(\bmod 8)≡5(mod8) that are not quadratic residues to each other, then n n nnn is a congruent number.
Theorem 7 (Burungale-Tian [11]). Conjecture B1 is true, namely among all square-free positive integers n 1 , 2 , 3 ( mod 8 ) n ≡ 1 , 2 , 3 ( mod 8 ) n-=1,2,3(mod8)n \equiv 1,2,3(\bmod 8)n≡1,2,3(mod8), the subset of n n nnn with L ( n ) 0 L ( n ) ≠ 0 L(n)!=0\mathscr{L}(n) \neq 0L(n)≠0 has density one. In particular, the density of noncongruent numbers among square-free positive integers 1 , 2 , 3 ( mod 8 ) ≡ 1 , 2 , 3 ( mod 8 ) -=1,2,3(mod8)\equiv 1,2,3(\bmod 8)≡1,2,3(mod8) is one.
Let S S SSS be the subset of positive square-free integers n 5 , 6 , 7 ( mod 8 ) n ≡ 5 , 6 , 7 ( mod 8 ) n-=5,6,7(mod8)n \equiv 5,6,7(\bmod 8)n≡5,6,7(mod8) so that dim F 2 Sel 2 ( E ( n ) / Q ) / E ( Q ) [ 2 ] = 1 dim F 2 ⁡ Sel 2 ⁡ E ( n ) / Q / E ( Q ) [ 2 ] = 1 dim_(F_(2))Sel_(2)(E^((n))//Q)//E(Q)[2]=1\operatorname{dim}_{\mathbb{F}_{2}} \operatorname{Sel}_{2}\left(E^{(n)} / \mathbb{Q}\right) / E(\mathbb{Q})[2]=1dimF2⁡Sel2⁡(E(n)/Q)/E(Q)[2]=1. By the results on distribution of 2-Selmer groups [27, 32,49], the density of S S SSS for all the positive square-free integers n 5 , 6 , 7 ( mod 8 ) n ≡ 5 , 6 , 7 ( mod 8 ) n-=5,6,7(mod8)n \equiv 5,6,7(\bmod 8)n≡5,6,7(mod8) is
2 j 1 ( 1 + 2 j ) 1 2 ∏ j ≥ 1   1 + 2 − j − 1 2prod_(j >= 1)(1+2^(-j))^(-1)2 \prod_{j \geq 1}\left(1+2^{-j}\right)^{-1}2∏j≥1(1+2−j)−1
Theorem 8 ( [ 47 , 50 , 51 ] ) 8 ( [ 47 , 50 , 51 ] ) 8([47,50,51])8([47,50,51])8([47,50,51]). There is a density- 2 3 2 3 (2)/(3)\frac{2}{3}23 subset of S S SSS so that the analytic rank of E ( n ) E ( n ) E^((n))E^{(n)}E(n) is one and the 2-part BSD formula holds. In particular, among all the square-free positive integers n 5 , 6 , 7 ( mod 8 ) n ≡ 5 , 6 , 7 ( mod 8 ) n-=5,6,7(mod8)n \equiv 5,6,7(\bmod 8)n≡5,6,7(mod8), the density of congruent numbers is greater than
4 3 j 1 ( 1 + 2 j ) 1 ( > 1 2 ) 4 3 ∏ j ≥ 1   1 + 2 − j − 1 > 1 2 (4)/(3)prod_(j >= 1)(1+2^(-j))^(-1)quad( > (1)/(2))\frac{4}{3} \prod_{j \geq 1}\left(1+2^{-j}\right)^{-1} \quad\left(>\frac{1}{2}\right)43∏j≥1(1+2−j)−1(>12)
The strategy of the proof of Theorems 6 and 8 (resp. Theorem 7) will be given in Section 2 (resp. Section 3).

2. HEEGNER POINT AND EXPLICIT GROSS-ZAGIER FORMULA

Heegner points and Gross-Zagier formula play an important role in the study of elliptic curves. The work of Yuan, Zhang, and Zhang [55] gives the general construction of Heegner points on Shimura curves over totally real fields and establishes the general GrossZagier formula. Some arithmetic applications require an explicit form of the formula such as that in [24]. In this section, we introduce the explicit Gross-Zagier formula from [15] and its application to CNP.
We assume as given:
  • A A AAA-an elliptic curve over Q Q Q\mathbb{Q}Q with conductor N N NNN,
  • K K KKK-an imaginary quadratic field with discriminant D D DDD,
  • χ χ chi\chiχ-a ring class character of K K KKK with conductor c c ccc, which can be viewed as a character on Gal ( H c / K ) Gal ⁡ H c / K Gal(H_(c)//K)\operatorname{Gal}\left(H_{c} / K\right)Gal⁡(Hc/K), where H c H c H_(c)H_{c}Hc is the ring class field of K K KKK with conductor c c ccc,
    characterized by the reciprocity law
t : Gal ( H c / K ) K × K ^ × / O ^ c × t : Gal ⁡ H c / K → ∼ K × ∖ K ^ × / O ^ c × t:Gal(H_(c)//K)rarr"∼"K^(xx)\\ hat(K)^(xx)// hat(O)_(c)^(xx)t: \operatorname{Gal}\left(H_{c} / K\right) \xrightarrow{\sim} K^{\times} \backslash \hat{K}^{\times} / \hat{\mathcal{O}}_{c}^{\times}t:Gal⁡(Hc/K)→∼K×∖K^×/O^c×
Here, O c = Z + c O K O c = Z + c O K O_(c)=Z+cO_(K)\mathcal{O}_{c}=\mathbb{Z}+c \mathcal{O}_{K}Oc=Z+cOK is an order of O K O K O_(K)\mathcal{O}_{K}OK. For any abelian group M M MMM, denote M ^ = M Z Z ^ M ^ = M ⊗ Z Z ^ hat(M)=Mox_(Z) hat(Z)\hat{M}=M \otimes_{\mathbb{Z}} \hat{\mathbb{Z}}M^=M⊗ZZ^ with Z ^ = p Z p Z ^ = ∏ p   Z p hat(Z)=prod_(p)Z_(p)\hat{\mathbb{Z}}=\prod_{p} \mathbb{Z}_{p}Z^=∏pZp.
Assume that the Rankin-Selberg L-series L ( s , A , χ ) L ( s , A , χ ) L(s,A,chi)L(s, A, \chi)L(s,A,χ) associated to ( A , χ ) ( A , χ ) (A,chi)(A, \chi)(A,χ) has sign -1 in its function equation.
In the following, we shall introduce the construction of the Heegner points and the explicit Gross-Zagier formula for ( A , χ ) ( A , χ ) (A,chi)(A, \chi)(A,χ) under the assumption ( c , N ) = 1 ( c , N ) = 1 (c,N)=1(c, N)=1(c,N)=1. Let B B BBB be the unique indefinite quaternion algebra over Q Q Q\mathbb{Q}Q whose ramified places are given by all p p ppp such that
ϵ p ( A , χ ) = χ p η p ( 1 ) ϵ p ( A , χ ) = − χ p η p ( − 1 ) epsilon_(p)(A,chi)=-chi_(p)eta_(p)(-1)\epsilon_{p}(A, \chi)=-\chi_{p} \eta_{p}(-1)ϵp(A,χ)=−χpηp(−1)
where ϵ p ( A , χ ) ϵ p ( A , χ ) epsilon_(p)(A,chi)\epsilon_{p}(A, \chi)ϵp(A,χ) is the local root number of L ( s , A , χ ) L ( s , A , χ ) L(s,A,chi)L(s, A, \chi)L(s,A,χ) at p p ppp and η p η p eta_(p)\eta_{p}ηp is the quadratic character of Q p × Q p × Q_(p)^(xx)\mathbb{Q}_{p}^{\times}Qp×associated to K p / Q p K p / Q p K_(p)//Q_(p)K_{p} / \mathbb{Q}_{p}Kp/Qp. In particular, there exists an embedding of K K KKK into B B BBB. Fix such an embedding once and for all. An order R R RRR of B B BBB is called admissible with respect to ( A , χ ) ( A , χ ) (A,chi)(A, \chi)(A,χ) if the discriminant of R R RRR is N N NNN and R K = O c R ∩ K = O c R nn K=O_(c)R \cap K=\mathcal{O}_{c}R∩K=Oc. Such an order exists and is unique up to conjugaction by K ^ × K ^ × hat(K)^(xx)\hat{K}^{\times}K^×. Fix such an admissible order R R RRR.
Denote by X R ^ × X R ^ × X_( hat(R)^(xx))X_{\hat{R}^{\times}}XR^×the Shimura curve over Q Q Q\mathbb{Q}Q associated to B B BBB of level R ^ × R ^ × hat(R)^(xx)\hat{R}^{\times}R^×. Under an isomorphism B ( R ) M 2 ( R ) B ( R ) ≃ M 2 ( R ) B(R)≃M_(2)(R)B(\mathbb{R}) \simeq M_{2}(\mathbb{R})B(R)≃M2(R), it has the following complex uniformization:
X R ^ × ( C ) = B × H ± × B ^ × / R ^ × { cusps } X R ^ × ( C ) = B × ∖ H ± × B ^ × / R ^ × ⊔ {  cusps  } X_( hat(R)^(xx))(C)=B^(xx)\\H^(+-)xx hat(B)^(xx)// hat(R)^(xx)⊔{" cusps "}X_{\hat{R}^{\times}}(\mathbb{C})=B^{\times} \backslash \mathscr{H}^{ \pm} \times \hat{B}^{\times} / \hat{R}^{\times} \sqcup\{\text { cusps }\}XR^×(C)=B×∖H±×B^×/R^×⊔{ cusps }
Denote by [ z , g ] R ^ × [ z , g ] R ^ × [z,g]_( hat(R)^(xx))[z, g]_{\hat{R}^{\times}}[z,g]R^×the image of ( z , g ) H ± × B ^ × ( z , g ) ∈ H ± × B ^ × (z,g)inH^(+-)xx hat(B)^(xx)(z, g) \in \mathscr{H}^{ \pm} \times \hat{B}^{\times}(z,g)∈H±×B^×in X R ^ × ( C ) X R ^ × ( C ) X_( hat(R)^(xx))(C)X_{\hat{R}^{\times}}(\mathbb{C})XR^×(C). Let ξ R ^ × Pic ( X R ^ × ) Q ξ R ^ × ∈ Pic ⁡ X R ^ × ⊗ Q xi_( hat(R)^(xx))in Pic(X_( hat(R)^(xx)))oxQ\xi_{\hat{R}^{\times}} \in \operatorname{Pic}\left(X_{\hat{R}^{\times}}\right) \otimes \mathbb{Q}ξR^×∈Pic⁡(XR^×)⊗Q be the normalized Hodge class with degree 1 on each connected component of X R ^ × , Q ¯ X R ^ × , Q ¯ X_( hat(R)^(xx), bar(Q))X_{\hat{R}^{\times}, \overline{\mathbb{Q}}}XR^×,Q¯ (see [ 55 [ 55 [55[55[55, SECTION 3.1.3]).
The following proposition follows from the modularity theorem and the JacquetLanglands correspondence.
Proposition 9 ([15, PROPOSITION 3.8]). Up to scalars, there is a unique nonconstant morphism f : X R ^ × f : X R ^ × → f:X_( hat(R)^(xx))rarrf: X_{\hat{R}^{\times}} \rightarrowf:XR^×→ A over Q Q Q\mathbb{Q}Q satisfying the following properties:
  • f f fff sends ξ R ^ × Î¾ R ^ × xi_( hat(R)^(xx))\xi_{\hat{R}^{\times}}ξR^×to the identity O O OOO of A A AAA in the sense that if ξ R ^ × Î¾ R ^ × xi_( hat(R)^(xx))\xi_{\hat{R}^{\times}}ξR^×is represented by a divisor n i x i ∑ n i x i sumn_(i)x_(i)\sum n_{i} x_{i}∑nixi on X R ^ × , Q ¯ X R ^ × , Q ¯ X_( hat(R)^(xx), bar(Q))X_{\hat{R}^{\times}, \overline{\mathbb{Q}}}XR^×,Q¯, then n i f ( x i ) = O ∑ n i f x i = O sumn_(i)f(x_(i))=O\sum n_{i} f\left(x_{i}\right)=O∑nif(xi)=O.
  • For each place p ( N , D ) p ∣ ( N , D ) p∣(N,D)p \mid(N, D)p∣(N,D),
T ϖ p f = χ p 1 ( ϖ p ) f T Ï– p f = χ p − 1 Ï– p f T_(Ï–_(p))f=chi_(p)^(-1)(Ï–_(p))fT_{\varpi_{p}} f=\chi_{p}^{-1}\left(\varpi_{p}\right) fTÏ–pf=χp−1(Ï–p)f
Here, T ϖ p T Ï– p T_(Ï–_(p))T_{\varpi_{p}}TÏ–p is the automorphism of X R ^ × X R ^ × X_( hat(R)^(xx))X_{\hat{R}^{\times}}XR^×, which on X R ^ × ( C ) X R ^ × ( C ) X_( hat(R)^(xx))(C)X_{\hat{R}^{\times}}(\mathbb{C})XR^×(C) is given by [ z , g ] R ^ × [ z , g ϖ p ] R ^ × [ z , g ] R ^ × ↦ z , g Ï– p R ^ × [z,g]_( hat(R)^(xx))|->[z,gÏ–_(p)]_( hat(R)^(xx))[z, g]_{\hat{R}^{\times}} \mapsto\left[z, g \varpi_{p}\right]_{\hat{R}^{\times}}[z,g]R^×↦[z,gÏ–p]R^×, with ϖ p K p × Ï– p ∈ K p × Ï–_(p)inK_(p)^(xx)\varpi_{p} \in K_{p}^{\times}Ï–p∈Kp×being any uniformizer of K p K p K_(p)K_{p}Kp.
Let z H z ∈ H z inHz \in \mathscr{H}z∈H be the unique point fixed by K × K × K^(xx)K^{\times}K×and let P = [ z , 1 ] R ^ × P = [ z , 1 ] R ^ × P=[z,1]_( hat(R)^(xx))P=[z, 1]_{\hat{R}^{\times}}P=[z,1]R^×. By the theory of complex multiplication, P P PPP is defined over the ring class field H c H c H_(c)H_{c}Hc of K K KKK with conductor c c ccc and the Galois action is given by
[ z , 1 ] R ^ × σ = [ z , t σ ] R ^ × , σ Gal ( H c / K ) [ z , 1 ] R ^ × σ = z , t σ R ^ × , σ ∈ Gal ⁡ H c / K [z,1]_( hat(R)^(xx))^(sigma)=[z,t_(sigma)]_( hat(R)^(xx)),quad sigma in Gal(H_(c)//K)[z, 1]_{\hat{R}^{\times}}^{\sigma}=\left[z, t_{\sigma}\right]_{\hat{R}^{\times}}, \quad \sigma \in \operatorname{Gal}\left(H_{c} / K\right)[z,1]R^×σ=[z,tσ]R^×,σ∈Gal⁡(Hc/K)
where σ t σ σ ↦ t σ sigma|->t_(sigma)\sigma \mapsto t_{\sigma}σ↦tσ is the reciprocity map.
Define the Heegner point
P χ ( f ) = σ Gal ( H c / K ) f ( P σ ) χ ( σ ) A ( H c ) Q ( χ ) P χ ( f ) = ∑ σ ∈ Gal ⁡ H c / K   f P σ χ ( σ ) ∈ A H c ⊗ Q ( χ ) P_(chi)(f)=sum_(sigma in Gal(H_(c)//K))f(P^(sigma))chi(sigma)in A(H_(c))oxQ(chi)P_{\chi}(f)=\sum_{\sigma \in \operatorname{Gal}\left(H_{c} / K\right)} f\left(P^{\sigma}\right) \chi(\sigma) \in A\left(H_{c}\right) \otimes \mathbb{Q}(\chi)Pχ(f)=∑σ∈Gal⁡(Hc/K)f(Pσ)χ(σ)∈A(Hc)⊗Q(χ)
Here Q ( χ ) Q ( χ ) Q(chi)\mathbb{Q}(\chi)Q(χ) is the field over Q Q Q\mathbb{Q}Q generated by image of χ χ chi\chiχ.
Theorem 10 ([15,55]). Assume ( N , c ) = 1 ( N , c ) = 1 (N,c)=1(N, c)=1(N,c)=1. Then
L ( 1 , A , χ ) = 2 μ ( N , D ) 8 π 2 ( ϕ , ϕ ) Γ 0 ( N ) u 2 | D c 2 | h ^ K ( P χ ( f ) ) deg f L ′ ( 1 , A , χ ) = 2 − μ ( N , D ) 8 Ï€ 2 ( Ï• , Ï• ) Γ 0 ( N ) u 2 D c 2 â‹… h ^ K P χ ( f ) deg ⁡ f L^(')(1,A,chi)=2^(-mu(N,D))(8pi^(2)(phi,phi)_(Gamma_(0)(N)))/(u^(2)sqrt(|Dc^(2)|))*( hat(h)_(K)(P_(chi)(f)))/(deg f)L^{\prime}(1, A, \chi)=2^{-\mu(N, D)} \frac{8 \pi^{2}(\phi, \phi)_{\Gamma_{0}(N)}}{u^{2} \sqrt{\left|D c^{2}\right|}} \cdot \frac{\hat{h}_{K}\left(P_{\chi}(f)\right)}{\operatorname{deg} f}L′(1,A,χ)=2−μ(N,D)8Ï€2(Ï•,Ï•)Γ0(N)u2|Dc2|â‹…h^K(Pχ(f))deg⁡f
Here, ϕ Ï• phi\phiÏ• is the newform associated to A A AAA with
( ϕ , ϕ ) Γ 0 ( N ) = Γ 0 ( N ) H | ϕ ( x + i y ) | 2 d x d y ( Ï• , Ï• ) Γ 0 ( N ) = ∬ Γ 0 ( N ) ∖ H   | Ï• ( x + i y ) | 2 d x d y (phi,phi)_(Gamma_(0)(N))=∬_(Gamma_(0)(N)\\H)|phi(x+iy)|^(2)dxdy(\phi, \phi)_{\Gamma_{0}(N)}=\iint_{\Gamma_{0}(N) \backslash \mathscr{H}}|\phi(x+i y)|^{2} d x d y(Ï•,Ï•)Γ0(N)=∬Γ0(N)∖H|Ï•(x+iy)|2dxdy
u = [ O c × : Z × ] , μ ( N , D ) u = O c × : Z × , μ ( N , D ) u=[O_(c)^(xx):Z^(xx)],mu(N,D)u=\left[\mathcal{O}_{c}^{\times}: \mathbb{Z}^{\times}\right], \mu(N, D)u=[Oc×:Z×],μ(N,D) is the number of common prime factors of N N NNN and D , h ^ K D , h ^ K D, hat(h)_(K)D, \hat{h}_{K}D,h^K is the Néron-Tate height on A over K K KKK, and deg ( f ) deg ⁡ ( f ) deg(f)\operatorname{deg}(f)deg⁡(f) is the degree of the morphism f f fff.

Remark 11.

(1) To compute Heegner points (if non-torsion) via CM theory and modular parameterization, one only gets an approximation. The precise computation can be carried out since one knows the height of Heegner point via the above formula (see [ 53 ] [ 53 ] [53][53][53] ).
(2) One may use different Heegner points to construct rational points on A A AAA by choosing different K K KKK. The case ( D , N ) 1 ( D , N ) ≠ 1 (D,N)!=1(D, N) \neq 1(D,N)≠1 sometimes provides points with smaller height. The above formula with ( D , N ) 1 ( D , N ) ≠ 1 (D,N)!=1(D, N) \neq 1(D,N)≠1 was conjectured by Gross and Hayashi in [25] and employed in [53] for the computation of rational points.
Some arithmetic problems lead to the situation ( c , N ) 1 ( c , N ) ≠ 1 (c,N)!=1(c, N) \neq 1(c,N)≠1. Consider the following: a nonzero rational number is called a cube sum if it is of form a 3 + b 3 a 3 + b 3 a^(3)+b^(3)a^{3}+b^{3}a3+b3 with a , b Q × a , b ∈ Q × a,b inQ^(xx)a, b \in \mathbb{Q}^{\times}a,b∈Q×. For any n Q × n ∈ Q × n inQ^(xx)n \in \mathbb{Q}^{\times}n∈Q×, consider the elliptic curve C n : x 3 + y 3 = 2 n C n : x 3 + y 3 = 2 n C_(n):x^(3)+y^(3)=2nC_{n}: x^{3}+y^{3}=2 nCn:x3+y3=2n. If n n nnn is not a cube, then 2 n 2 n 2n2 n2n is a cube sum if and only if the rank of C n ( Q ) C n ( Q ) C_(n)(Q)C_{n}(\mathbb{Q})Cn(Q) is positive.
Theorem 12 ([16]). For any odd integer k 1 k ≥ 1 k >= 1k \geq 1k≥1, there exist infinitely many cube-free odd integers n n nnn with exactly k k kkk distinct prime factors such that
rank Z C n ( Q ) = 1 = ord s = 1 L ( s , C n ) rank Z ⁡ C n ( Q ) = 1 = ord s = 1 ⁡ L s , C n rank_(Z)C_(n)(Q)=1=ord_(s=1)L(s,C_(n))\operatorname{rank}_{\mathbb{Z}} C_{n}(\mathbb{Q})=1=\operatorname{ord}_{s=1} L\left(s, C_{n}\right)rankZ⁡Cn(Q)=1=ords=1⁡L(s,Cn)
Here, a certain Heegner point is considered for the pair ( A , χ ) ( A , χ ) (A,chi)(A, \chi)(A,χ) where A = X 0 ( 36 ) A = X 0 ( 36 ) A=X_(0)(36)A=X_{0}(36)A=X0(36) : x 2 = y 3 + 1 x 2 = y 3 + 1 x^(2)=y^(3)+1x^{2}=y^{3}+1x2=y3+1 is an isogeny to C 1 C 1 C_(1)C_{1}C1 and χ χ chi\chiχ is a certain cubic ring class character over the imaginary quadratic field Q ( 3 ) Q ( − 3 ) Q(sqrt(-3))\mathbb{Q}(\sqrt{-3})Q(−3) with conductor 3 n 3 n ∗ 3n^(**)3 n^{*}3n∗ where n n ∗ n^(**)n^{*}n∗ is the product of prime factors in n n nnn. In particular, the pair ( A , χ ) ( A , χ ) (A,chi)(A, \chi)(A,χ) has joint ramification at the prime 3 .
In fact, the explicit Gross-Zagier formula is proved for any pair ( π , χ ) ( Ï€ , χ ) (pi,chi)(\pi, \chi)(Ï€,χ) where
  • π Ï€ pi\piÏ€ is a cuspidal automorphic representation on G L 2 G L 2 GL_(2)\mathrm{GL}_{2}GL2 over a totally real field F F FFF with central character ω π ω Ï€ omega_(pi)\omega_{\pi}ωπ, discrete series of weight 2 at all archimedean places,
  • χ : K × K ^ × C × Ï‡ : K × ∖ K ^ × → C × chi:K^(xx)\\ hat(K)^(xx)rarrC^(xx)\chi: K^{\times} \backslash \hat{K}^{\times} \rightarrow \mathbb{C}^{\times}χ:K×∖K^×→C×is a character of finite order for a totally imaginary quadratic extension K K KKK over F F FFF such that
(i) ω π χ | A F × = 1 ω Ï€ â‹… χ A F × = 1 omega_(pi)*chi|_(A_(F)^(xx))=1\left.\omega_{\pi} \cdot \chi\right|_{\mathbb{A}_{F}^{\times}}=1ωπ⋅χ|AF×=1,
(ii) the root number of the Rankin-Selberg L-series L ( s , π , χ ) L ( s , Ï€ , χ ) L(s,pi,chi)L(s, \pi, \chi)L(s,Ï€,χ) is -1 .
Based on the work of Yuan-Zhang-Zhang [55], the above explicit formula is established via generalizing Gross-Prasad local test vector theory.
The relevant problem in local harmonic analysis is the following. Let B B B\mathscr{B}B be a quaternion algebra over a local field F F F\mathscr{F}F with a quadratic sub- F F F\mathcal{F}F-algebra K K K\mathcal{K}K. Let π Ï€ pi\piÏ€ be an irreducible smooth admissible representation on B × B × B^(xx)\mathscr{B}^{\times}B×which is of infinite dimension if B B B\mathscr{B}B is split. Let χ χ chi\chiχ

the functional space
P ( π , χ ) := Hom K × ( π , χ 1 ) P ( Ï€ , χ ) := Hom K × Ï€ , χ − 1 P(pi,chi):=Hom_(K)xx(pi,chi^(-1))\mathcal{P}(\pi, \chi):=\operatorname{Hom}_{\mathcal{K}} \times\left(\pi, \chi^{-1}\right)P(Ï€,χ):=HomK×(Ï€,χ−1)
In general, dim C P ( π , χ ) 1 dim C ⁡ P ( Ï€ , χ ) ≤ 1 dim_(C)P(pi,chi) <= 1\operatorname{dim}_{\mathbb{C}} \mathcal{P}(\pi, \chi) \leq 1dimC⁡P(Ï€,χ)≤1. In the case P ( π , χ ) 0 P ( Ï€ , χ ) ≠ 0 P(pi,chi)!=0\mathcal{P}(\pi, \chi) \neq 0P(Ï€,χ)≠0, a vector φ φ varphi\varphiφ is called a test vector for ( π , χ ) ( Ï€ , χ ) (pi,chi)(\pi, \chi)(Ï€,χ) if ( φ ) 0 â„“ ( φ ) ≠ 0 â„“(varphi)!=0\ell(\varphi) \neq 0â„“(φ)≠0 for any nonzero P ( π , χ ) â„“ ∈ P ( Ï€ , χ ) â„“inP(pi,chi)\ell \in \mathcal{P}(\pi, \chi)ℓ∈P(Ï€,χ).
Moreover, for an unitary ( π , χ ) , P ( π , χ ) ( Ï€ , χ ) , P ( Ï€ , χ ) (pi,chi),P(pi,chi)(\pi, \chi), \mathscr{P}(\pi, \chi)(Ï€,χ),P(Ï€,χ) is nonzero if and only if the bilinear form α P ( π , χ ) P ( π ¯ , χ ¯ ) α ∈ P ( Ï€ , χ ) ⊗ P ( Ï€ ¯ , χ ¯ ) alpha inP(pi,chi)oxP( bar(pi), bar(chi))\alpha \in \mathcal{P}(\pi, \chi) \otimes \mathscr{P}(\bar{\pi}, \bar{\chi})α∈P(Ï€,χ)⊗P(π¯,χ¯) defined as the toric integral of matrix coefficients
α ( φ 1 φ 2 ) = F × K × π ( t ) φ 1 , φ 2 χ ( t ) d t , φ 1 π , φ 2 π ¯ Î± φ 1 ⊗ φ 2 = ∫ F × ∖ K ×   Ï€ ( t ) φ 1 , φ 2 χ ( t ) d t , φ 1 ∈ Ï€ , φ 2 ∈ Ï€ ¯ alpha(varphi_(1)oxvarphi_(2))=int_(F^(xx)\\Kxx)(:pi(t)varphi_(1),varphi_(2):)chi(t)dt,quadvarphi_(1)in pi,varphi_(2)in bar(pi)\alpha\left(\varphi_{1} \otimes \varphi_{2}\right)=\int_{\mathcal{F}^{\times} \backslash \mathcal{K} \times}\left\langle\pi(t) \varphi_{1}, \varphi_{2}\right\rangle \chi(t) d t, \quad \varphi_{1} \in \pi, \varphi_{2} \in \bar{\pi}α(φ1⊗φ2)=∫F×∖K×⟨π(t)φ1,φ2⟩χ(t)dt,φ1∈π,φ2∈π¯
is nonzero. Here, π ¯ Ï€ ¯ bar(pi)\bar{\pi}π¯ (resp. χ ¯ χ ¯ bar(chi)\bar{\chi}χ¯ ) is the complex conjugate of π Ï€ pi\piÏ€ (resp. χ χ chi\chiχ ) and , ⟨ â‹… , â‹… ⟩ (:*,*:)\langle\cdot, \cdot\rangle⟨⋅,⋅⟩ is a nondegenerate invariant pairing on π π ¯ Ï€ ⊗ Ï€ ¯ pi ox bar(pi)\pi \otimes \bar{\pi}π⊗π¯. In particular, if P ( π , χ ) 0 , φ P ( Ï€ , χ ) ≠ 0 , φ P(pi,chi)!=0,varphi\mathcal{P}(\pi, \chi) \neq 0, \varphiP(Ï€,χ)≠0,φ is a test vector for ( π , χ ) ( Ï€ , χ ) (pi,chi)(\pi, \chi)(Ï€,χ) if and only if α ( φ , φ ¯ ) 0 α ( φ , φ ¯ ) ≠ 0 alpha(varphi, bar(varphi))!=0\alpha(\varphi, \bar{\varphi}) \neq 0α(φ,φ¯)≠0.
For any pair ( π , χ ) ( Ï€ , χ ) (pi,chi)(\pi, \chi)(Ï€,χ) as above, in [15] we find an admissible order R R R\mathcal{R}R for ( π , χ ) ( Ï€ , χ ) (pi,chi)(\pi, \chi)(Ï€,χ) which is unique up to K × K × K^(xx)\mathcal{K}^{\times}K×-conjugacy. The invariant subspace π R × Ï€ R × pi^(R^(xx))\pi^{\mathcal{R}^{\times}}Ï€R×of π Ï€ pi\piÏ€ by R × R × R^(xx)\mathcal{R}^{\times}R×is at most of dimension 2. By studying the toric integral α α alpha\alphaα, there is a line in π R × Ï€ R × pi^(R^(xx))\pi^{\mathcal{R}^{\times}}Ï€R×containing test vectors for ( π , χ ) ( Ï€ , χ ) (pi,chi)(\pi, \chi)(Ï€,χ).
Our explicit Gross-Zagier formula satisfies the following properties: First, the test vector only depends on the local type π v , χ v Ï€ v , χ v pi_(v),chi_(v)\pi_{v}, \chi_{v}Ï€v,χv, for v v vvv dividing the conductor of π Ï€ pi\piÏ€. It is useful when considering horizontal variation (quadratic twist, for example), see [7,13], or vertical variation (in Iwasawa theory) of the character χ χ chi\chiχ. We also have a so-called S S SSS-version formula which says that for a different choice of a pure tensor test vector, for example, at a finite set of places S S SSS, the new explicit formula can be obtained by modifying the original one by local computations at S S SSS, for example, see [16].
In the rest of this section, we sketch a proofs of Theorems 6 and 8. In Heegner's work, the point y n y n y_(n)y_{n}yn is not 2-divisible. In [50,51], Heegner's results were generalized to many prime factors by induction on 2-divisibility of Heegner points via the Waldspurger and GrossZagier formulas.
For E : y 2 = x 3 x , K = Q ( n ) , n 5 , 6 , 7 E : y 2 = x 3 − x , K = Q ( − n ) , n ≡ 5 , 6 , 7 E:y^(2)=x^(3)-x,K=Q(sqrt(-n)),n-=5,6,7E: y^{2}=x^{3}-x, K=\mathbb{Q}(\sqrt{-n}), n \equiv 5,6,7E:y2=x3−x,K=Q(−n),n≡5,6,7 positive square-free, the explicit Gross-Zagier formula for ( E , K ) ( E , K ) (E,K)(E, K)(E,K) gives
h ^ Q ( y n ) = 2 a L ( 1 , E ( n ) ) Ω E ( n ) h ^ Q y n = 2 a L ′ 1 , E ( n ) Ω E ( n ) hat(h)_(Q)(y_(n))=2^(a)(L^(')(1,E^((n))))/(Omega_(E^((n))))\hat{h}_{\mathbb{Q}}\left(y_{n}\right)=2^{a} \frac{L^{\prime}\left(1, E^{(n)}\right)}{\Omega_{E^{(n)}}}h^Q(yn)=2aL′(1,E(n))ΩE(n)
where a = 1 , 0 , 1 a = 1 , 0 , 1 a=1,0,1a=1,0,1a=1,0,1 in the case n 5 , 6 , 7 ( mod 8 ) n ≡ 5 , 6 , 7 ( mod 8 ) n-=5,6,7(mod8)n \equiv 5,6,7(\bmod 8)n≡5,6,7(mod8), respectively. Now if y n y n y_(n)y_{n}yn is nontorsion, then the BSD conjecture becomes
[ E ( n ) ( Q ) / E ( n ) ( Q ) t o r : Z y n ] = 2 μ ( n ) 1 # ⨿ ( E ( n ) / Q ) E ( n ) ( Q ) / E ( n ) ( Q ) t o r : Z â‹… y n = 2 μ ( n ) − 1 # ⨿ E ( n ) / Q [E^((n))(Q)//E^((n))(Q)_(tor):Z*y_(n)]=2^(mu(n)-1)sqrt#⨿(E^((n))//Q)\left[E^{(n)}(\mathbb{Q}) / E^{(n)}(\mathbb{Q})_{\mathrm{tor}}: \mathbb{Z} \cdot y_{n}\right]=2^{\mu(n)-1} \sqrt{\# \amalg\left(E^{(n)} / \mathbb{Q}\right)}[E(n)(Q)/E(n)(Q)tor:Zâ‹…yn]=2μ(n)−1#⨿(E(n)/Q)
where μ ( n ) μ ( n ) mu(n)\mu(n)μ(n) is the number of odd prime factors of n n nnn. If n n nnn is a prime then the 2-part of the BSD conjecture is equivalent to 2-indivisibility of y n y n y_(n)y_{n}yn, this is exactly Heegner's case. As μ ( n ) μ ( n ) mu(n)\mu(n)μ(n) becomes large, the 2 divisibility of y n y n y_(n)y_{n}yn becomes high and the original Heegner's argument does not work directly. Whenever dim F 2 Sel 2 ( E ( n ) / Q ) / E ( n ) ( Q ) [ 2 ] = 1 dim F 2 ⁡ Sel 2 ⁡ E ( n ) / Q / E ( n ) ( Q ) [ 2 ] = 1 dim_(F_(2))Sel_(2)(E^((n))//Q)//E^((n))(Q)[2]=1\operatorname{dim}_{\mathbb{F}_{2}} \operatorname{Sel}_{2}\left(E^{(n)} / \mathbb{Q}\right) / E^{(n)}(\mathbb{Q})[2]=1dimF2⁡Sel2⁡(E(n)/Q)/E(n)(Q)[2]=1, the 2-divisibility of y n y n y_(n)y_{n}yn fully comes from Tamagawa numbers. The 2-divisibility can be proved via induction; to do this, one employs various relations between different Heegner points.
We employ the induction method (see [50]) in the case n 5 ( mod 8 ) n ≡ 5 ( mod 8 ) n-=5(mod8)n \equiv 5(\bmod 8)n≡5(mod8) with all prime factors 1 ( mod 4 ) ≡ 1 ( mod 4 ) -=1(mod4)\equiv 1(\bmod 4)≡1(mod4). Let z n := f ( τ n ) z n := f Ï„ n z_(n):=f(tau_(n))z_{n}:=f\left(\tau_{n}\right)zn:=f(Ï„n) and let y n y n y_(n)y_{n}yn be the Heegner points as in the Section 1. Denote by H H HHH the Hilbert class field of K = Q ( n ) K = Q ( − n ) K=Q(sqrt(-n))K=\mathbb{Q}(\sqrt{-n})K=Q(−n) and let H 0 H H 0 ⊂ H H_(0)sub HH_{0} \subset HH0⊂H be the genus subfield determined by Gal ( H 0 / K ) 2 C l ( K ) Gal ⁡ H 0 / K ≃ 2 C l ( K ) Gal(H_(0)//K)≃2Cl(K)\operatorname{Gal}\left(H_{0} / K\right) \simeq 2 \mathrm{Cl}(K)Gal⁡(H0/K)≃2Cl(K). For each d n d ∣ n d∣nd \mid nd∣n with the same above property as n n nnn, let y 0 = tr H / H 0 z n y 0 = tr H / H 0 ⁡ z n y_(0)=tr_(H//H_(0))z_(n)y_{0}=\operatorname{tr}_{H / H_{0}} z_{n}y0=trH/H0⁡zn and y d , 0 = tr H / K ( d ) z n y d , 0 = tr H / K ( − d ) ⁡ z n y_(d,0)=tr_(H//K(sqrt(-d)))z_(n)y_{d, 0}=\operatorname{tr}_{H / K(\sqrt{-d})} z_{n}yd,0=trH/K(−d)⁡zn. Then these points satisfy the following relation:
y n + 1 d n , d n , d 5 ( mod 8 ) y d , 0 = 2 μ ( n ) y 0 ( mod E [ 2 ] ) y n + ∑ 1 ≤ d ∣ n , d ≠ n , d ≡ 5 ( mod 8 )   y d , 0 = 2 μ ( n ) y 0 ( mod E [ 2 ] ) y_(n)+sum_({:1 <= d∣n","d!=n","d-=5(mod8):})y_(d,0)=2^(mu(n))y_(0)quad(mod E[2])y_{n}+\sum_{\substack{1 \leq d \mid n, d \neq n, d \equiv 5(\bmod 8)}} y_{d, 0}=2^{\mu(n)} y_{0} \quad(\bmod E[2])yn+∑1≤d∣n,d≠n,d≡5(mod8)yd,0=2μ(n)y0(modE[2])
Furthermore, the Gross-Zagier formula implies that whenever y d , 0 y d , 0 y_(d,0)y_{d, 0}yd,0 is nontorsion, both y d , 0 y d , 0 y_(d,0)y_{d, 0}yd,0 and y d y d y_(d)y_{d}yd lie in the one-dimensional space E ( Q ( d ) ) c = 1 Q E ( Q ( − d ) ) c = − 1 ⊗ Q E(Q(sqrt(-d)))^(c=-1)oxQE(\mathbb{Q}(\sqrt{-d}))^{c=-1} \otimes \mathbb{Q}E(Q(−d))c=−1⊗Q and
[ y d , 0 : y d ] 2 = L a l g ( 1 , E ( n / d ) ) L a l g ( 1 , E ) , where L a l g ( 1 , E ( n / d ) ) = L ( 1 , E ( n / d ) ) Ω E ( n / d ) y d , 0 : y d 2 = L a l g 1 , E ( n / d ) L a l g ( 1 , E ) ,  where  L a l g 1 , E ( n / d ) = L 1 , E ( n / d ) Ω E ( n / d ) [y_(d,0):y_(d)]^(2)=(L^(alg)(1,E^((n//d))))/(L^(alg)(1,E)),quad" where "L^(alg)(1,E^((n//d)))=(L(1,E^((n//d))))/(Omega_(E^((n//d))))\left[y_{d, 0}: y_{d}\right]^{2}=\frac{L^{\mathrm{alg}}\left(1, E^{(n / d)}\right)}{L^{\mathrm{alg}}(1, E)}, \quad \text { where } L^{\mathrm{alg}}\left(1, E^{(n / d)}\right)=\frac{L\left(1, E^{(n / d)}\right)}{\Omega_{E^{(n / d)}}}[yd,0:yd]2=Lalg(1,E(n/d))Lalg(1,E), where Lalg(1,E(n/d))=L(1,E(n/d))ΩE(n/d)
By induction on the 2 divisibility of y d y d y_(d)y_{d}yd and 2 divisibility of L als ( 1 , E ( n / d ) ) L a l g ( 1 , E ) L als  1 , E ( n / d ) L a l g ( 1 , E ) (L^("als ")(1,E^((n//d))))/(L^(alg)(1,E))\frac{L^{\text {als }}\left(1, E^{(n / d)}\right)}{L^{\mathrm{alg}}(1, E)}Lals (1,E(n/d))Lalg(1,E), one gets the following 2 divisibility of y n y n y_(n)y_{n}yn whenever C l ( K ) C l ( K ) Cl(K)\mathrm{Cl}(K)Cl(K) has no element of order 4 :
y n ( 2 μ ( n ) 1 E ( K ) c = 1 + E ( K ) t o r ) ( 2 μ ( n ) E ( K ) c = 1 + E ( K ) t o r ) y n ∈ 2 μ ( n ) − 1 E ( K ) c = − 1 + E ( K ) t o r ∖ 2 μ ( n ) E ( K ) c = − 1 + E ( K ) t o r y_(n)in(2^(mu(n)-1)E(K)^(c=-1)+E(K)_(tor))\\(2^(mu(n))E(K)^(c=-1)+E(K)_(tor))y_{n} \in\left(2^{\mu(n)-1} E(K)^{c=-1}+E(K)_{\mathrm{tor}}\right) \backslash\left(2^{\mu(n)} E(K)^{c=-1}+E(K)_{\mathrm{tor}}\right)yn∈(2μ(n)−1E(K)c=−1+E(K)tor)∖(2μ(n)E(K)c=−1+E(K)tor)
Thus Theorem 6 follows.
The above induction argument was improved in [51] to handle the general case. For a positive integer d d ddd, let g ( d ) = # 2 C l ( Q ( d ) ) g ( d ) = # 2 C l ( Q ( − d ) ) g(d)=#2Cl(Q(sqrt(-d)))g(d)=\# 2 \mathrm{Cl}(\mathbb{Q}(\sqrt{-d}))g(d)=#2Cl(Q(−d)) be the genus class number of Q ( d ) Q ( − d ) Q(sqrt(-d))\mathbb{Q}(\sqrt{-d})Q(−d). For n 5 , 6 , 7 ( mod 8 ) n ≡ 5 , 6 , 7 ( mod 8 ) n-=5,6,7(mod8)n \equiv 5,6,7(\bmod 8)n≡5,6,7(mod8), let
where a ( n ) = 0 a ( n ) = 0 a(n)=0a(n)=0a(n)=0 if n n nnn is even and a ( n ) = 1 a ( n ) = 1 a(n)=1a(n)=1a(n)=1 if n n nnn is odd.
Then the BSD conjecture for E ( n ) E ( n ) E^((n))E^{(n)}E(n) is equivalent to L ( n ) 2 = # ⨿ ( E ( n ) / Q ) L ( n ) 2 = # ⨿ E ( n ) / Q L(n)^(2)=#⨿(E^((n))//Q)\mathscr{L}(n)^{2}=\# \amalg\left(E^{(n)} / \mathbb{Q}\right)L(n)2=#⨿(E(n)/Q) whenever L ( 1 , E ( n ) ) 0 L ′ 1 , E ( n ) ≠ 0 L^(')(1,E^((n)))!=0L^{\prime}\left(1, E^{(n)}\right) \neq 0L′(1,E(n))≠0. We have the following criterion for the 2-indivisibility of L ( n ) L ( n ) L(n)\mathscr{L}(n)L(n) :
Theorem 13 ([51]). For n 5 , 6 , 7 ( mod 8 ) n ≡ 5 , 6 , 7 ( mod 8 ) n-=5,6,7(mod8)n \equiv 5,6,7(\bmod 8)n≡5,6,7(mod8) positive square-free, L ( n ) L ( n ) L(n)\mathscr{L}(n)L(n) is an integer and 2 ρ ( n ) L ( n ) 2 − ρ ( n ) L ( n ) 2^(-rho(n))L(n)2^{-\rho(n)} \mathscr{L}(n)2−ρ(n)L(n) is odd if
where ρ ( n ) ρ ( n ) rho(n)\rho(n)ρ(n) is a positive integer (defined in [51]) arising from an isogeny between E ( n ) E ( n ) E^((n))E^{(n)}E(n) and 2 n y 2 = x 3 + x 2 n y 2 = x 3 + x 2ny^(2)=x^(3)+x2 n y^{2}=x^{3}+x2ny2=x3+x.
Let
s ( n ) = dim F 2 Sel 2 ( E ( n ) / Q ) / E ( n ) ( Q ) [ 2 ] = rank Z E ( n ) ( Q ) + dim F 2 ⨿ ( E ( n ) / Q ) [ 2 ] s ( n ) = dim F 2 ⁡ Sel 2 ⁡ E ( n ) / Q / E ( n ) ( Q ) [ 2 ] = rank Z ⁡ E ( n ) ( Q ) + dim F 2 ⨿ E ( n ) / Q [ 2 ] {:[s(n)=dim_(F_(2))Sel_(2)(E^((n))//Q)//E^((n))(Q)[2]],[=rank_(Z)E^((n))(Q)+dim_(F_(2))⨿(E^((n))//Q)[2]]:}\begin{aligned} s(n) & =\operatorname{dim}_{\mathbb{F}_{2}} \operatorname{Sel}_{2}\left(E^{(n)} / \mathbb{Q}\right) / E^{(n)}(\mathbb{Q})[2] \\ & =\operatorname{rank}_{\mathbb{Z}} E^{(n)}(\mathbb{Q})+\operatorname{dim}_{\mathbb{F}_{2}} \amalg\left(E^{(n)} / \mathbb{Q}\right)[2] \end{aligned}s(n)=dimF2⁡Sel2⁡(E(n)/Q)/E(n)(Q)[2]=rankZ⁡E(n)(Q)+dimF2⨿(E(n)/Q)[2]
Consider the following sets for i = 5 , 6 , 7 i = 5 , 6 , 7 i=5,6,7i=5,6,7i=5,6,7 :
  • Σ i Σ i Sigma_(i)\Sigma_{i}Σi-the set of all square-free positive integers n i ( mod 8 ) n ≡ i ( mod 8 ) n-=i(mod8)n \equiv i(\bmod 8)n≡i(mod8),
  • Σ i Σ i Σ i ′ ⊂ Σ i Sigma_(i)^(')subSigma_(i)\Sigma_{i}^{\prime} \subset \Sigma_{i}Σi′⊂Σi-the subset of n n nnn with s ( n ) = 1 s ( n ) = 1 s(n)=1s(n)=1s(n)=1,
  • Σ i Σ i Σ i ′ ′ ⊂ Σ i Sigma_(i)^('')subSigma_(i)\Sigma_{i}^{\prime \prime} \subset \Sigma_{i}Σi′′⊂Σi-the subset of n n nnn satisfying the conditions in the Theorem 13.
Theorem 14 (Heath-Brown [27], Swinnerton-Dyer [49], Kane [32]). The density of Σ i Σ i ′ Sigma_(i)^(')\Sigma_{i}^{\prime}Σi′ in Σ i Σ i Sigma_(i)\Sigma_{i}Σi is
2 k = 1 ( 1 + 2 k ) 1 = 0.8388 2 ∏ k = 1 ∞   1 + 2 − k − 1 = 0.8388 … 2prod_(k=1)^(oo)(1+2^(-k))^(-1)=0.8388 dots2 \prod_{k=1}^{\infty}\left(1+2^{-k}\right)^{-1}=0.8388 \ldots2∏k=1∞(1+2−k)−1=0.8388…
Theorem 15 (Smith [47]). The set Σ i Σ i ′ ′ Sigma_(i)^('')\Sigma_{i}^{\prime \prime}Σi′′ is contained in Σ i Σ i ′ Sigma_(i)^(')\Sigma_{i}^{\prime}Σi′ with density 3 4 , 1 2 , 3 4 3 4 , 1 2 , 3 4 (3)/(4),(1)/(2),(3)/(4)\frac{3}{4}, \frac{1}{2}, \frac{3}{4}34,12,34 for i = 5 , 6 , 7 i = 5 , 6 , 7 i=5,6,7i=5,6,7i=5,6,7, respectively.
Observe that Theorem 8 is a consequence of Theorems 13, 14, and 15.

3. SELMER GROUPS: P-CONVERSE AND DISTRIBUTION

The n n nnn-Selmer group for an elliptic curve A A AAA over a number F F FFF is defined by
Sel n ( A / F ) = ker ( H 1 ( F , A [ n ] ) v H 1 ( F v , A ) ) Sel n ⁡ ( A / F ) = ker ⁡ H 1 ( F , A [ n ] ) → ∏ v   H 1 F v , A Sel_(n)(A//F)=ker(H^(1)(F,A[n])rarrprod_(v)H^(1)(F_(v),A))\operatorname{Sel}_{n}(A / F)=\operatorname{ker}\left(H^{1}(F, A[n]) \rightarrow \prod_{v} H^{1}\left(F_{v}, A\right)\right)Seln⁡(A/F)=ker⁡(H1(F,A[n])→∏vH1(Fv,A))
and fits into the short exact sequence
0 A ( F ) / n A ( F ) Sel n ( A / F ) ⨿ ( A / F ) [ n ] 0 0 → A ( F ) / n A ( F ) → Sel n ⁡ ( A / F ) → ⨿ ( A / F ) [ n ] → 0 0rarr A(F)//nA(F)rarrSel_(n)(A//F)rarr⨿(A//F)[n]rarr00 \rightarrow A(F) / n A(F) \rightarrow \operatorname{Sel}_{n}(A / F) \rightarrow \amalg(A / F)[n] \rightarrow 00→A(F)/nA(F)→Seln⁡(A/F)→⨿(A/F)[n]→0
The group Hom ( Sel p ( A / F ) , Q p / Z p ) Hom ⁡ Sel p ⁡ ∞ ( A / F ) , Q p / Z p Hom(Sel_(p)oo(A//F),Q_(p)//Z_(p))\operatorname{Hom}\left(\operatorname{Sel}_{p} \infty(A / F), \mathbb{Q}_{p} / \mathbb{Z}_{p}\right)Hom⁡(Selp⁡∞(A/F),Qp/Zp) is known to be a finitely generated Z p Z p Z_(p)\mathbb{Z}_{p}Zp-module, its rank of free part is called p p ∞ p^(oo)p^{\infty}p∞-Selmer corank of A A AAA, denoted by corank Z p Sel p ( A / F ) Z p Sel p ∞ ⁡ ( A / F ) Z_(p)Sel_(p^(oo))(A//F)\mathbb{Z}_{p} \operatorname{Sel}_{p^{\infty}}(A / F)ZpSelp∞⁡(A/F).
Conjecture 16 (BSD, reformulation). Let A / F A / F A//FA / FA/F be an elliptic curve over a number field, r Z > 0 r ∈ Z > 0 r inZ_( > 0)r \in \mathbb{Z}_{>0}r∈Z>0, and p p ppp be a prime. Then the following are equivalent:
(1) ord s = 1 L ( s , A / F ) = r ord s = 1 ⁡ L ( s , A / F ) = r ord_(s=1)L(s,A//F)=r\operatorname{ord}_{s=1} L(s, A / F)=rords=1⁡L(s,A/F)=r,
(2) rank Z A ( F ) = r rank Z ⁡ A ( F ) = r rank_(Z)A(F)=r\operatorname{rank}_{\mathbb{Z}} A(F)=rrankZ⁡A(F)=r and ⨿ ( A / F ) ⨿ ( A / F ) ⨿(A//F)\amalg(A / F)⨿(A/F) is finite,
(3) corank Z p Sel p ( A / F ) = r corank Z p ⁡ Sel p ⁡ ∞ ( A / F ) = r corank_(Z_(p))Sel_(p)oo(A//F)=r\operatorname{corank}_{\mathbb{Z}_{p}} \operatorname{Sel}_{p} \infty(A / F)=rcorankZp⁡Selp⁡∞(A/F)=r.
Notation ( p p ppp-converse). The implication
corank Z p Sel p ( A / F ) = r ord s = 1 L ( s , A / F ) = r corank Z p ⁡ Sel p ∞ ⁡ ( A / F ) = r ⟹ ord s = 1 ⁡ L ( s , A / F ) = r corank_(Z_(p))Sel_(p)^(oo)(A//F)=r Longrightarroword_(s=1)L(s,A//F)=r\operatorname{corank}_{\mathbb{Z}_{p}} \operatorname{Sel}_{p}^{\infty}(A / F)=r \Longrightarrow \operatorname{ord}_{s=1} L(s, A / F)=rcorankZp⁡Selp∞⁡(A/F)=r⟹ords=1⁡L(s,A/F)=r
is referred to as rank r r rrr p-converse.
We can also consider a Selmer variant of Goldfeld's Conjecture 3. The following was conjectured by Bhargava-Kane-Lenstra-Poonen-Rains.
Conjecture 17 ([4]). Let F F FFF be a global field, p p ppp be a prime, and G G GGG a finite symplectic p p ppp-group. If all elliptic curves A over F F FFF are ordered by height, then for r = 0 , 1 r = 0 , 1 r=0,1r=0,1r=0,1 we have
Prob ( Sel p ( A / F ) ( Q p / Z p ) r G ) = 1 2 ( # G ) 1 r # Sp ( G ) i r ( 1 p 1 2 i ) Prob ⁡ Sel p ⁡ ∞ ( A / F ) ≃ Q p / Z p r ⊕ G = 1 2 ⋅ ( # G ) 1 − r # Sp ⁡ ( G ) ⋅ ∏ i ≥ r   1 − p 1 − 2 i Prob(Sel_(p)oo(A//F)≃(Q_(p)//Z_(p))^(r)o+G)=(1)/(2)*((#G)^(1-r))/(#Sp(G))*prod_(i >= r)(1-p^(1-2i))\operatorname{Prob}\left(\operatorname{Sel}_{p} \infty(A / F) \simeq\left(\mathbb{Q}_{p} / \mathbb{Z}_{p}\right)^{r} \oplus G\right)=\frac{1}{2} \cdot \frac{(\# G)^{1-r}}{\# \operatorname{Sp}(G)} \cdot \prod_{i \geq r}\left(1-p^{1-2 i}\right)Prob⁡(Selp⁡∞(A/F)≃(Qp/Zp)r⊕G)=12⋅(#G)1−r#Sp⁡(G)⋅∏i≥r(1−p1−2i)
In particular, the density of rank 0 (or 1 ) elliptic curves over F F FFF is 1 2 1 2 (1)/(2)\frac{1}{2}12.

3.1. Distribution of Selmer groups and Goldfeld conjecture

The following Smith's result shows that even for a quadratic twist family, the distribution follows the same pattern as the above conjecture.
Theorem 18 (Smith [48]). Let A / Q A / Q A//QA / \mathbb{Q}A/Q be an elliptic curve satisfying A A AAA has full rational 2 torsion, but no rational cyclic subgroup of order 4 . Then, among the quadratic twists A ( d ) A ( d ) A^((d))A^{(d)}A(d) of A, a distribution law as in Conjecture 17 holds for p = 2 p = 2 p=2p=2p=2.
In particular, among all quadratic twists A ( d ) A ( d ) A^((d))A^{(d)}A(d) with sign +1 (resp. -1 ), there is a subset of density one with corank Z 2 Sel 2 ( A ( d ) / Q ) = 0 Z 2 Sel 2 ⁡ A ( d ) / Q = 0 Z_(2)Sel_(2)(A^((d))//Q)=0\mathbb{Z}_{2} \operatorname{Sel}_{2}\left(A^{(d)} / \mathbb{Q}\right)=0Z2Sel2⁡(A(d)/Q)=0 (resp. 1 ).
Remark 19. Smith's work is based on the following results of Heath-Brown, SwinnertonDyer, and Kane on distribution of 2-Selmer groups, which is the first step to understand the distribution of 2 2 ∞ 2^(oo)2^{\infty}2∞-Selmer groups.
Theorem 20 ([27,32,49]). Let A / Q A / Q A//QA / \mathbb{Q}A/Q be an elliptic curve satisfying
  • A has full rational 2-torsion, but no rational cyclic subgroup of order 4 .
Then for r Z 0 r ∈ Z ≥ 0 r inZ_( >= 0)r \in \mathbb{Z}_{\geq 0}r∈Z≥0, among the quadratic twists A ( d ) A ( d ) A^((d))A^{(d)}A(d) of A A AAA,
Prob ( dim F 2 Sel 2 ( A ( d ) / Q ) / A ( d ) ( Q ) [ 2 ] = r ) = j = 0 ( 1 + 2 j ) 1 i = 1 r 2 2 i 1 Prob ⁡ dim F 2 ⁡ Sel 2 ⁡ A ( d ) / Q / A ( d ) ( Q ) [ 2 ] = r = ∏ j = 0 ∞   1 + 2 − j − 1 ∏ i = 1 r   2 2 i − 1 Prob(dim_(F_(2))Sel_(2)(A^((d))//Q)//A^((d))(Q)[2]=r)=prod_(j=0)^(oo)(1+2^(-j))^(-1)prod_(i=1)^(r)(2)/(2^(i)-1)\operatorname{Prob}\left(\operatorname{dim}_{\mathbb{F}_{2}} \operatorname{Sel}_{2}\left(A^{(d)} / \mathbb{Q}\right) / A^{(d)}(\mathbb{Q})[2]=r\right)=\prod_{j=0}^{\infty}\left(1+2^{-j}\right)^{-1} \prod_{i=1}^{r} \frac{2}{2^{i}-1}Prob⁡(dimF2⁡Sel2⁡(A(d)/Q)/A(d)(Q)[2]=r)=∏j=0∞(1+2−j)−1∏i=1r22i−1
In general, for a quadratic twist family of elliptic curves over Q Q Q\mathbb{Q}Q, its distribution of 2-Selmer groups may exhibit new behavior. For example, the quadratic twist family of Tiling
number elliptic curves has
A ( d ) : d y 2 = x ( x 1 ) ( x + 3 ) with A ( 1 ) ( Q ) tor Z / 2 Z × Z / 4 Z A ( d ) : d y 2 = x ( x − 1 ) ( x + 3 )  with  A ( 1 ) ( Q ) tor  ≅ Z / 2 Z × Z / 4 Z A^((d)):dy^(2)=x(x-1)(x+3)quad" with "A^((1))(Q)_("tor ")~=Z//2ZxxZ//4ZA^{(d)}: d y^{2}=x(x-1)(x+3) \quad \text { with } A^{(1)}(\mathbb{Q})_{\text {tor }} \cong \mathbb{Z} / 2 \mathbb{Z} \times \mathbb{Z} / 4 \mathbb{Z}A(d):dy2=x(x−1)(x+3) with A(1)(Q)tor ≅Z/2Z×Z/4Z
Perhaps surprisingly, in light of the presence of such rational 4-torsion, the distribution of 2-Selmer groups no longer seems to be as in Theorem 20. For example, if d 1 d ≠ 1 d!=1d \neq 1d≠1, d 1 ( mod 12 ) d ≡ 1 ( mod 12 ) d-=1(mod 12)d \equiv 1(\bmod 12)d≡1(mod12) is positive square-free, then
dim F 2 Sel 2 ( A ( d ) / Q ) / A ( d ) ( Q ) [ 2 ] 2 dim F 2 ⁡ Sel 2 ⁡ A ( − d ) / Q / A ( − d ) ( Q ) [ 2 ] ≥ 2 dim_(F_(2))Sel_(2)(A^((-d))//Q)//A^((-d))(Q)[2] >= 2\operatorname{dim}_{\mathbb{F}_{2}} \operatorname{Sel}_{2}\left(A^{(-d)} / \mathbb{Q}\right) / A^{(-d)}(\mathbb{Q})[2] \geq 2dimF2⁡Sel2⁡(A(−d)/Q)/A(−d)(Q)[2]≥2
A preliminary study suggests that for such elliptic curves, the distribution of 2Selmer groups may look more like that of the 4-ranks of ideal class groups of the underlying imaginary quadratic fields.
Theorem 21 ([22]). Let A A AAA be the elliptic curve y 2 = x ( x 1 ) ( x + 3 ) y 2 = x ( x − 1 ) ( x + 3 ) y^(2)=x(x-1)(x+3)y^{2}=x(x-1)(x+3)y2=x(x−1)(x+3). Among the set of positive square-free integers d 7 ( mod 24 ) ( d ≡ 7 ( mod 24 ) ( d-=7(mod 24)(d \equiv 7(\bmod 24)(d≡7(mod24)( resp. d 3 ( mod 24 ) ) d ≡ 3 ( mod 24 ) ) d-=3(mod 24))d \equiv 3(\bmod 24))d≡3(mod24)), the subset of d d ddd such that both of A ( ± d ) A ( ± d ) A^((+-d))A^{( \pm d)}A(±d) have Sel 2 ( A ( ± d ) / Q ) / A ( ± d ) ( Q ) [ 2 ] Sel 2 ⁡ A ( ± d ) / Q / A ( ± d ) ( Q ) [ 2 ] Sel_(2)(A^((+-d))//Q)//A^((+-d))(Q)[2]\operatorname{Sel}_{2}\left(A^{( \pm d)} / \mathbb{Q}\right) / A^{( \pm d)}(\mathbb{Q})[2]Sel2⁡(A(±d)/Q)/A(±d)(Q)[2] trivial has density 1 2 i = 1 ( 1 2 i ) > 1 2 ∏ i = 1 ∞   1 − 2 − i > (1)/(2)prod_(i=1)^(oo)(1-2^(-i)) >\frac{1}{2} \prod_{i=1}^{\infty}\left(1-2^{-i}\right)>12∏i=1∞(1−2−i)> 14.4 % 14.4 % 14.4%14.4 \%14.4% (resp. of density i = 1 ( 1 2 i ) > 28.8 % ∏ i = 1 ∞   1 − 2 − i > 28.8 % prod_(i=1)^(oo)(1-2^(-i)) > 28.8%\prod_{i=1}^{\infty}\left(1-2^{-i}\right)>28.8 \%∏i=1∞(1−2−i)>28.8% ).
In general, for r Z 0 r ∈ Z ≥ 0 r inZ_( >= 0)r \in \mathbb{Z}_{\geq 0}r∈Z≥0, for the set of positive square-free integers d 3 ( mod 24 ) d ≡ 3 ( mod 24 ) d-=3(mod 24)d \equiv 3(\bmod 24)d≡3(mod24),
Prob ( dim F 2 Sel 2 ( A ( d ) / Q ) / A ( d ) ( Q ) [ 2 ] = 2 r ) Prob ⁡ dim F 2 ⁡ Sel 2 ⁡ A ( d ) / Q / A ( d ) ( Q ) [ 2 ] = 2 r Prob(dim_(F_(2))Sel_(2)(A^((d))//Q)//A^((d))(Q)[2]=2r)\operatorname{Prob}\left(\operatorname{dim}_{\mathbb{F}_{2}} \operatorname{Sel}_{2}\left(A^{(d)} / \mathbb{Q}\right) / A^{(d)}(\mathbb{Q})[2]=2 r\right)Prob⁡(dimF2⁡Sel2⁡(A(d)/Q)/A(d)(Q)[2]=2r)
= ( k = 0 r 2 ( r + k ) ( 3 r + 3 k 1 ) / 2 i = 1 r + k ( 1 2 i ) 2 i = 0 2 k 1 ( 2 r + k i 1 ) i = 1 k 4 i 1 4 i 1 ) i = 1 ( 1 2 i ) = ∑ k = 0 r   2 − ( r + k ) ( 3 r + 3 k − 1 ) / 2 ∏ i = 1 r + k   1 − 2 − i − 2 ∏ i = 0 2 k − 1   2 r + k − i − 1 ∏ i = 1 k   4 i − 1 4 i − 1 ⋅ ∏ i = 1 ∞   1 − 2 − i =(sum_(k=0)^(r)2^(-(r+k)(3r+3k-1)//2)prod_(i=1)^(r+k)(1-2^(-i))^(-2)prod_(i=0)^(2k-1)(2^(r+k-i)-1)prod_(i=1)^(k)(4^(i-1))/(4^(i)-1))*prod_(i=1)^(oo)(1-2^(-i))=\left(\sum_{k=0}^{r} 2^{-(r+k)(3 r+3 k-1) / 2} \prod_{i=1}^{r+k}\left(1-2^{-i}\right)^{-2} \prod_{i=0}^{2 k-1}\left(2^{r+k-i}-1\right) \prod_{i=1}^{k} \frac{4^{i-1}}{4^{i}-1}\right) \cdot \prod_{i=1}^{\infty}\left(1-2^{-i}\right)=(∑k=0r2−(r+k)(3r+3k−1)/2∏i=1r+k(1−2−i)−2∏i=02k−1(2r+k−i−1)∏i=1k4i−14i−1)⋅∏i=1∞(1−2−i)
An approach to Goldfeld conjecture. The Goldfeld conjecture for a quadratic twist family of elliptic curves over Q Q Q\mathbb{Q}Q is a consequence of the following steps:
(1) Distribution of p p ∞ p^(oo)p^{\infty}p∞-Selmer groups in the quadratic twist family, which should be a certain variant of general distribution law for all elliptic curves in [4].
(2) The rank zero and rank one p p ppp-converse.
Proof of Theorem 7. It is a direct consequence of Tunnell's work on quadratic twist L-values of congruent elliptic curves [52], Theorem 18 of Smith on distribution of 2 2 ∞ 2^(oo)2^{\infty}2∞-Selmer groups, and Theorem 22 below on the rank zero p p ppp-converse for C M C M CM\mathrm{CM}CM elliptic curves for p = 2 p = 2 p=2p=2p=2.

3.2. Recent progress: p p ppp-converse

In the remaining part of this section, we discuss the p p ppp-converse theorem in the C M C M CM\mathrm{CM}CM case. For a few other p p ppp-converse theorems, see [6,8-10]. Fix a prime p p ppp.
Theorem 22 (Rubin [43,44], Burungale-Tian [11]). Let A / Q A / Q A//QA / \mathbb{Q}A/Q be a CM elliptic curve. Then,
corank Z p Sel p ( A / Q ) = 0 ord s = 1 L ( s , A / Q ) = 0 corank Z p ⁡ Sel p ⁡ ∞ ( A / Q ) = 0 ⟹ ord s = 1 ⁡ L ( s , A / Q ) = 0 corank_(Z_(p))Sel_(p)oo(A//Q)=0Longrightarroword_(s=1)L(s,A//Q)=0\operatorname{corank}_{\mathbb{Z}_{p}} \operatorname{Sel}_{p} \infty(A / \mathbb{Q})=0 \Longrightarrow \operatorname{ord}_{s=1} L(s, A / \mathbb{Q})=0corankZp⁡Selp⁡∞(A/Q)=0⟹ords=1⁡L(s,A/Q)=0
Remark 23. Assume that A / Q A / Q A//QA / \mathbb{Q}A/Q has C M C M CMC MCM by K K KKK and p # O K × p ∤ # O K × p∤#O_(K)^(xx)p \nmid \# \mathcal{O}_{K}^{\times}p∤#OK×. Then the above theorem is due to Rubin [ 43 , 44 ] [ 43 , 44 ] [43,44][43,44][43,44].
Remark 24. Skinner-Urban [46] established the rank zero p-converse for certain elliptic curves over Q Q Q\mathbb{Q}Q without CM.
Theorem 25 (W. Zhang [56], Skinner [45], Castella-Wan [17]). Let A/QQ be a non-CM elliptic curve and p 3 p ≥ 3 p >= 3p \geq 3p≥3. Then,
corank Z p Sel p ( A / Q ) = 1 ord s = 1 L ( s , A / Q ) = 1 corank Z p ⁡ Sel p ⁡ ∞ ( A / Q ) = 1 ⟹ ord s = 1 ⁡ L ( s , A / Q ) = 1 corank_(Z_(p))Sel_(p)oo(A//Q)=1Longrightarroword_(s=1)L(s,A//Q)=1\operatorname{corank}_{\mathbb{Z}_{p}} \operatorname{Sel}_{p} \infty(A / \mathbb{Q})=1 \Longrightarrow \operatorname{ord}_{s=1} L(s, A / \mathbb{Q})=1corankZp⁡Selp⁡∞(A/Q)=1⟹ords=1⁡L(s,A/Q)=1
under certain assumptions.
Their methods essentially excludes the CM case. For CM elliptic curves:
Theorem 26 (Burungale-Tian [12], Burungale-Skinner-Tian [8]). Let A be a CM elliptic curve over Q Q Q\mathbb{Q}Q and p 6 N A p ∤ 6 N A p∤6N_(A)p \nmid 6 N_{A}p∤6NA a prime. Then
corank Z p Sel p ( A / Q ) = 1 ord s = 1 L ( s , A / Q ) = 1 corank Z p ⁡ Sel p ⁡ ∞ ( A / Q ) = 1 ⟹ ord s = 1 ⁡ L ( s , A / Q ) = 1 corank_(Z_(p))Sel_(p)oo(A//Q)=1Longrightarroword_(s=1)L(s,A//Q)=1\operatorname{corank}_{\mathbb{Z}_{p}} \operatorname{Sel}_{p} \infty(A / \mathbb{Q})=1 \Longrightarrow \operatorname{ord}_{s=1} L(s, A / \mathbb{Q})=1corankZp⁡Selp⁡∞(A/Q)=1⟹ords=1⁡L(s,A/Q)=1

3.2.1. Rank zero CM p p ppp-converse

We outline the proof of Theorem 22. Unconventionally for the CM elliptic curves, this approach is based on Kato's main conjecture [33], which we recall now.
Let f S k ( Γ 0 ( N ) ) f ∈ S k Γ 0 ( N ) f inS_(k)(Gamma_(0)(N))f \in S_{k}\left(\Gamma_{0}(N)\right)f∈Sk(Γ0(N)) be an elliptic newform of even weight k 2 k ≥ 2 k >= 2k \geq 2k≥2, level Γ 0 ( N ) Γ 0 ( N ) Gamma_(0)(N)\Gamma_{0}(N)Γ0(N), and Hecke field F F FFF. Fix an embedding ι p : Q Q ¯ p ι p : Q → Q ¯ p iota_(p):Qrarr bar(Q)_(p)\iota_{p}: \mathbb{Q} \rightarrow \overline{\mathbb{Q}}_{p}ιp:Q→Q¯p. Let λ λ lambda\lambdaλ be the place of F F FFF induced by ι p , F λ ι p , F λ iota_(p),F_(lambda)\iota_{p}, F_{\lambda}ιp,Fλ be the completion of F F FFF at λ λ lambda\lambdaλ, and O λ O λ O_(lambda)O_{\lambda}Oλ the integer ring. Let V F λ ( f ) V F λ ( f ) V_(F_(lambda))(f)V_{F_{\lambda}}(f)VFλ(f) be the two-dimensional representation of G Q G Q G_(Q)G_{\mathbb{Q}}GQ over F λ F λ F_(lambda)F_{\lambda}Fλ associated to f f fff. We first introduce the related Iwasawa cohomology. For n Z 0 n ∈ Z ≥ 0 n inZ_( >= 0)n \in \mathbb{Z}_{\geq 0}n∈Z≥0, let
G n = Gal ( Q ( ζ p n ) / Q ) , G = lim n G n = Gal ( Q ( ζ p ) / Q ) G n = Gal ⁡ Q ζ p n / Q , G ∞ = lim n   G n = Gal ⁡ Q ζ p ∞ / Q G_(n)=Gal(Q(zeta_(p^(n)))//Q),quadG_(oo)=lim _(n)G_(n)=Gal(Q(zeta_(p^(oo)))//Q)G_{n}=\operatorname{Gal}\left(\mathbb{Q}\left(\zeta_{p^{n}}\right) / \mathbb{Q}\right), \quad G_{\infty}=\underset{n}{\lim } G_{n}=\operatorname{Gal}\left(\mathbb{Q}\left(\zeta_{p^{\infty}}\right) / \mathbb{Q}\right)Gn=Gal⁡(Q(ζpn)/Q),G∞=limnGn=Gal⁡(Q(ζp∞)/Q)
Let Λ = O λ [ [ G ] ] Λ = O λ G ∞ Lambda=O_(lambda)[[G_(oo)]]\Lambda=O_{\lambda}\left[\left[G_{\infty}\right]\right]Λ=Oλ[[G∞]] be a two-dimensional complete semilocal ring. For q Z 0 q ∈ Z ≥ 0 q inZ_( >= 0)q \in \mathbb{Z}_{\geq 0}q∈Z≥0, consider the Λ Q p = Λ Q Λ Q p = Λ ⊗ Q Lambda_(Q_(p))=Lambda oxQ\Lambda_{\mathbb{Q}_{p}}=\Lambda \otimes \mathbb{Q}ΛQp=Λ⊗Q-module
H q ( V F λ ( f ) ) = lim n H q ( Z [ ζ p n , 1 / p ] , T ) Q H q V F λ ( f ) = lim n   H q Z ζ p n , 1 / p , T ⊗ Q H^(q)(V_(F_(lambda))(f))=lim _(n)H^(q)(Z[zeta_(p^(n)),1//p],T)oxQ\mathbb{H}^{q}\left(V_{F_{\lambda}}(f)\right)=\underset{n}{\lim } H^{q}\left(\mathbb{Z}\left[\zeta_{p^{n}}, 1 / p\right], T\right) \otimes \mathbb{Q}Hq(VFλ(f))=limnHq(Z[ζpn,1/p],T)⊗Q
where T V F λ ( f ) T ⊂ V F λ ( f ) T subV_(F_(lambda))(f)T \subset V_{F_{\lambda}}(f)T⊂VFλ(f) is any G Q G Q G_(Q)G_{\mathbb{Q}}GQ-stable O λ O λ O_(lambda)O_{\lambda}Oλ-lattice and H q H q H^(q)H^{q}Hq denotes the étale cohomology. The following holds [33]:
(1) H 2 ( V F λ ( f ) ) H 2 V F λ ( f ) H^(2)(V_(F_(lambda))(f))\mathbb{H}^{2}\left(V_{F_{\lambda}}(f)\right)H2(VFλ(f)) is a torsion Λ Q p Λ Q p Lambda_(Q_(p))\Lambda_{\mathbb{Q}_{p}}ΛQp-module, and
(2) H 1 ( V F λ ( f ) ) H 1 V F λ ( f ) H^(1)(V_(F_(lambda))(f))\mathbb{H}^{1}\left(V_{F_{\lambda}}(f)\right)H1(VFλ(f)) is a free Λ Q p Λ Q p Lambda_(Q_(p))\Lambda_{\mathbb{Q}_{p}}ΛQp-module of rank one.
Now we introduce the submodule of H 1 ( V F λ ( f ) ) H 1 V F λ ( f ) H^(1)(V_(F_(lambda))(f))\mathbb{H}^{1}\left(V_{F_{\lambda}}(f)\right)H1(VFλ(f)) generated by Beilinson-Kato elements. We have the following existence of zeta elements for the p p ppp-adic Galois representation corresponding to an elliptic newform [33, Ñ‚HM. 12.5]:
(1) There exists a nonzero F λ F λ F_(lambda)F_{\lambda}Fλ-linear morphism
V F λ ( f ) H 1 ( V F λ ( f ) ) ; γ z γ ( f ) V F λ ( f ) → H 1 V F λ ( f ) ; γ ↦ z γ ( f ) V_(F_(lambda))(f)rarrH^(1)(V_(F_(lambda))(f));gamma|->z_(gamma)(f)V_{F_{\lambda}}(f) \rightarrow \mathbb{H}^{1}\left(V_{F_{\lambda}}(f)\right) ; \gamma \mapsto z_{\gamma}(f)VFλ(f)→H1(VFλ(f));γ↦zγ(f)
(2) Let Z ( f ) Z ( f ) Z(f)Z(f)Z(f) be the Λ Q p Λ Q p Lambda_(Q_(p))\Lambda_{\mathbb{Q}_{p}}ΛQp-submodule of H 1 ( V F λ ( f ) ) H 1 V F λ ( f ) H^(1)(V_(F_(lambda))(f))\mathbb{H}^{1}\left(V_{F_{\lambda}}(f)\right)H1(VFλ(f)) generated by z γ ( f ) z γ ( f ) z_(gamma)(f)z_{\gamma}(f)zγ(f) for all γ V F λ ( f ) γ ∈ V F λ ( f ) gamma inV_(F_(lambda))(f)\gamma \in V_{F_{\lambda}}(f)γ∈VFλ(f). Then H 1 ( V F λ ( f ) ) / Z ( f ) H 1 V F λ ( f ) / Z ( f ) H^(1)(V_(F_(lambda))(f))//Z(f)\mathbb{H}^{1}\left(V_{F_{\lambda}}(f)\right) / Z(f)H1(VFλ(f))/Z(f) is a torsion Λ Q p Λ Q p Lambda_(Q_(p))\Lambda_{\mathbb{Q}_{p}}ΛQp-module.
Remark 27. For a characterizing property of the morphism γ z γ ( f ) γ ↦ z γ ( f ) gamma|->z_(gamma)(f)\gamma \mapsto z_{\gamma}(f)γ↦zγ(f) in terms of the underlying critical L-values, we refer to [33, тнм. 12.5 (1)].
Conjecture 28 (Kato's Main Conjecture [33]). The following equality of ideals holds in Λ Q p Λ Q p Lambda_(Q_(p))\Lambda_{\mathbb{Q}_{p}}ΛQp :
Char ( H 2 ( V F λ ( f ) ) ) = Char ( H 1 ( V F λ ( f ) ) / Z ( f ) ) Char ⁡ H 2 V F λ ( f ) = Char ⁡ H 1 V F λ ( f ) / Z ( f ) Char(H^(2)(V_(F_(lambda))(f)))=Char(H^(1)(V_(F_(lambda))(f))//Z(f))\operatorname{Char}\left(\mathbb{H}^{2}\left(V_{F_{\lambda}}(f)\right)\right)=\operatorname{Char}\left(\mathbb{H}^{1}\left(V_{F_{\lambda}}(f)\right) / Z(f)\right)Char⁡(H2(VFλ(f)))=Char⁡(H1(VFλ(f))/Z(f))
Theorem 29 (Burungale-Tian [11]). Kato's main conjecture holds for any CM modular form f f fff and any prime p p ppp.
As observed by Kato [33], the CM case of Kato's main conjecture is closely related to an equivariant main conjecture for the underlying imaginary quadratic field. This is based on an intrinsic relation between the Beilinson-Kato elements and elliptic units.
As a consequence of Theorem 29, we have the following result, which implies Theorem 22 .
Theorem 30 ([11]). Assume that f f fff is C M C M CMC MCM. Let H f 1 ( Q , V F λ ( f ) ( k / 2 ) ) H f 1 Q , V F λ ( f ) ( k / 2 ) H_(f)^(1)(Q,V_(F_(lambda))(f)(k//2))H_{\mathrm{f}}^{1}\left(\mathbb{Q}, V_{F_{\lambda}}(f)(k / 2)\right)Hf1(Q,VFλ(f)(k/2)) be the corresponding Bloch-Kato Selmer group (see Kato [33]). Then,
H f 1 ( Q , V F λ ( f ) ( k / 2 ) ) = 0 ord s = k / 2 L ( s , f ) = 0 H f 1 Q , V F λ ( f ) ( k / 2 ) = 0 ⟹ ord s = k / 2 ⁡ L ( s , f ) = 0 H_(f)^(1)(Q,V_(F_(lambda))(f)(k//2))=0Longrightarroword_(s=k//2)L(s,f)=0H_{\mathrm{f}}^{1}\left(\mathbb{Q}, V_{F_{\lambda}}(f)(k / 2)\right)=0 \Longrightarrow \operatorname{ord}_{s=k / 2} L(s, f)=0Hf1(Q,VFλ(f)(k/2))=0⟹ords=k/2⁡L(s,f)=0

3.3. Rank one CM p p p\boldsymbol{p}p-converse

In the following, we focus on the proof of the rank one CM p p ppp-converse theorem for ordinary primes p p ppp. The key is an auxiliary Heegner point main conjecture (HPMC, for short).
Classically, HPMC is only formulated for pairs ( A , K ) A , K ′ (A,K^('))\left(A, K^{\prime}\right)(A,K′) where A / Q A / Q A//QA / \mathbb{Q}A/Q is an elliptic curve and K K ′ K^(')K^{\prime}K′ is an imaginary quadratic field satisfying the Heegner hypothesis. To show the rank 1 -converse for a CM elliptic curve, we utilize a certain anticyclotomic Iwasawa theory over the CM field. The key is to construct relevant Heegner points for auxiliary RankinSelberg pairs, and consider the underlying HPMC.
Let A / Q A / Q A//QA / \mathbb{Q}A/Q be a CM elliptic curve with C M C M CM\mathrm{CM}CM by K K KKK and with p p ∞ p^(oo)p^{\infty}p∞-Selmer corank one. Let λ λ lambda\lambdaλ be the associated Hecke character over K K KKK and θ λ θ λ theta_(lambda)\theta_{\lambda}θλ the corresponding theta series.
Lemma 31. There exists a finite order Hecke character χ χ chi\chiχ over K K KKK such that L ( 1 , λ / χ χ ) 0 L 1 , λ ∗ / χ ∗ â‹… χ ≠ 0 L(1,lambda^(**)//chi^(**)*chi)!=0L\left(1, \lambda^{*} / \chi^{*} \cdot \chi\right) \neq 0L(1,λ∗/χ∗⋅χ)≠0, where ∗ ***∗ is the involution given by nontrivial automorphism of K K KKK, so that the L L LLL-function for the Rankin pair ( f := θ λ / χ , χ ) f := θ λ / χ , χ (f:=theta_(lambda//chi),chi)\left(f:=\theta_{\lambda / \chi}, \chi\right)(f:=θλ/χ,χ),
L ( s , f × χ ) = L ( s , λ ) L ( s , λ / χ χ ) L ( s , f × χ ) = L ( s , λ ) L s , λ ∗ / χ ∗ â‹… χ L(s,f xx chi)=L(s,lambda)L(s,lambda^(**)//chi^(**)*chi)L(s, f \times \chi)=L(s, \lambda) L\left(s, \lambda^{*} / \chi^{*} \cdot \chi\right)L(s,f×χ)=L(s,λ)L(s,λ∗/χ∗⋅χ)
has sign -1 and the same vanishing order at the center as L ( s , λ ) = L ( s , A / Q ) L ( s , λ ) = L ( s , A / Q ) L(s,lambda)=L(s,A//Q)L(s, \lambda)=L(s, A / \mathbb{Q})L(s,λ)=L(s,A/Q).
We have the relevant Heegner point P 0 B ( K ) Q P 0 ∈ B ( K ) ⊗ Q P_(0)in B(K)oxQP_{0} \in B(K) \otimes \mathbb{Q}P0∈B(K)⊗Q on the abelian variety B := A f × χ B := A f × χ B:=A_(f xx chi)B:=A_{f \times \chi}B:=Af×χ associated to the pair ( f = θ λ / χ , χ ) f = θ λ / χ , χ (f=theta_(lambda//chi),chi)\left(f=\theta_{\lambda / \chi}, \chi\right)(f=θλ/χ,χ). The Gross-Zagier formula of YuanZhang-Zhang [55] implies that 0 P 0 B ( K ) Q 0 ≠ P 0 ∈ B ( K ) ⊗ Q 0!=P_(0)in B(K)oxQ0 \neq P_{0} \in B(K) \otimes \mathbb{Q}0≠P0∈B(K)⊗Q if and only if ord s = 1 L ( s , f × χ ) = 1 s = 1 L ( s , f × χ ) = 1 _(s=1)L(s,f xx chi)=1{ }_{s=1} L(s, f \times \chi)=1s=1L(s,f×χ)=1.
Note that p p ppp is split in K K KKK. Let K / K K ∞ / K K_(oo)//KK_{\infty} / KK∞/K be the anticyclotomic extension with Galois group Γ Z p Γ ≅ Z p Gamma~=Z_(p)\Gamma \cong \mathbb{Z}_{p}Γ≅Zp. For each n 1 n ≥ 1 n >= 1n \geq 1n≥1, let K n K K n ⊂ K ∞ K_(n)subK_(oo)K_{n} \subset K_{\infty}Kn⊂K∞ be the degree p n p n p^(n)p^{n}pn subextension over K K KKK.
One can construct a family of norm compatible Heegner points P n B ( K n ) P n ∈ B K n P_(n)in B(K_(n))P_{n} \in B\left(K_{n}\right)Pn∈B(Kn). Denote by Λ = O p [ [ Γ ] ] Λ = O p [ [ Γ ] ] Lambda=O_(p)[[Gamma]]\Lambda=\mathcal{O}_{\mathfrak{p}}[[\Gamma]]Λ=Op[[Γ]] and Λ Q p = Λ Q Λ Q p = Λ ⊗ Q Lambda_(Q_(p))=Lambda oxQ\Lambda_{\mathbb{Q}_{p}}=\Lambda \otimes \mathbb{Q}ΛQp=Λ⊗Q. Here O O O\mathcal{O}O is the endomorphism ring of B B BBB (viewed as a subring of Q ¯ ) , p p Q ¯ ) , p ∣ p bar(Q)),p∣p\overline{\mathbb{Q}}), \mathfrak{p} \mid pQ¯),p∣p the prime ideal of O O O\mathcal{O}O induced by ι p ι p iota_(p)\iota_{p}ιp, and O p O p O_(p)\mathcal{O}_{\mathfrak{p}}Op the completion of O O O\mathcal{O}O at p p ppp.
Proposition 32. The Λ Q p Λ Q p Lambda_(Q_(p))\Lambda_{\mathbb{Q}_{p}}ΛQp-modules
S ( B / K ) := ( lim n m lim m Sel p m ( B / K n ) ) Q p , X ( B / K ) := ( lim n lim m Sel p m ( B / K n ) ) Q p S B / K ∞ := lim n   m lim m ←   Sel p m ⁡ B / K n Q p , X B / K ∞ := ( lim n lim m   Sel p m ⁡ B / K n Q p ∨ S(B//K_(oo)):=(lim _(n)_(m)lim_(m^( larr))Sel_(p^(m))(B//K_(n)))_(Q_(p)),quad X(B//K_(oo)):=(lim_(n)lim _(m)Sel_(p^(m))(B//K_(n)))_(Q_(p))^(vv)\left.S\left(B / K_{\infty}\right):=\left(\underset{n}{\lim } \underset{m}{ } \lim _{\overleftarrow{m}} \operatorname{Sel}_{\mathfrak{p}^{m}}\left(B / K_{n}\right)\right)_{\mathbb{Q}_{p}}, \quad X\left(B / K_{\infty}\right):=\underset{n}{(\lim } \underset{m}{\lim } \operatorname{Sel}_{\mathfrak{p}^{m}}\left(B / K_{n}\right)\right)_{\mathbb{Q}_{p}}^{\vee}S(B/K∞):=(limnmlimm←Selpm⁡(B/Kn))Qp,X(B/K∞):=(limnlimmSelpm⁡(B/Kn))Qp∨
are finitely generated Λ Q p Λ Q p Lambda_(Q_(p))\Lambda_{\mathbb{Q}_{p}}ΛQp-modules of rank one. Moreover, the element
κ = ( P n ) S ( B / K ) κ = P n ∈ S B / K ∞ kappa=(P_(n))in S(B//K_(oo))\kappa=\left(P_{n}\right) \in S\left(B / K_{\infty}\right)κ=(Pn)∈S(B/K∞)
is not Λ Q p Λ Q p Lambda_(Q_(p))\Lambda_{\mathbb{Q}_{p}}ΛQp-torsion so that S ( B / K ) / Λ Q p κ S B / K ∞ / Λ Q p â‹… κ S(B//K_(oo))//Lambda_(Q_(p))*kappaS\left(B / K_{\infty}\right) / \Lambda_{\mathbb{Q}_{p}} \cdot \kappaS(B/K∞)/ΛQp⋅κ is a finitely generated torsion Λ Q p -module. Λ Q p -module.  Lambda_(Q_(p)"-module. ")\Lambda_{\mathbb{Q}_{p} \text {-module. }}ΛQp-module. .
Conjecture 33 (HPMC). With the above notations,
( Char ( S ( B / K ) / ( κ ) ) ) 2 = Char ( X ( B / K ) t o r ) Char ⁡ S B / K ∞ / ( κ ) 2 = Char ⁡ X B / K ∞ t o r (Char(S(B//K_(oo))//(kappa)))^(2)=Char(X(B//K_(oo))_(tor))\left(\operatorname{Char}\left(S\left(B / K_{\infty}\right) /(\kappa)\right)\right)^{2}=\operatorname{Char}\left(X\left(B / K_{\infty}\right)_{\mathrm{tor}}\right)(Char⁡(S(B/K∞)/(κ)))2=Char⁡(X(B/K∞)tor)
The control theorem gives
# ( S ( B / K ) / ( κ ) ) Γ < 0 P 0 B ( K ) corank O p Sel p ( B / K ) = 1 # ( X ( B / K ) tor ) Γ < # S B / K ∞ / ( κ ) Γ < ∞ ⇒ 0 ≠ P 0 ∈ B ( K ) corank O p ⁡ Sel p ⁡ ∞ ( B / K ) = 1 ⇒ # X B / K ∞ tor Γ < ∞ {:[#(S(B//K_(oo))//(kappa))_(Gamma) < oo=>0!=P_(0)in B(K)],[corank_(O_(p))Sel_(p)oo(B//K)=1=>#(X(B//K_(oo))_(tor))_(Gamma) < oo]:}\begin{gathered} \#\left(S\left(B / K_{\infty}\right) /(\kappa)\right)_{\Gamma}<\infty \Rightarrow 0 \neq P_{0} \in B(K) \\ \operatorname{corank}_{\mathcal{O}_{\mathfrak{p}}} \operatorname{Sel}_{\mathfrak{p}} \infty(B / K)=1 \Rightarrow \#\left(X\left(B / K_{\infty}\right)_{\operatorname{tor}}\right)_{\Gamma}<\infty \end{gathered}#(S(B/K∞)/(κ))Γ<∞⇒0≠P0∈B(K)corankOp⁡Selp⁡∞(B/K)=1⇒#(X(B/K∞)tor)Γ<∞
The rank one p p ppp-converse in the CM case is a consequence of HPMC. In fact, by descent,
(*) # ( X ( B / K ) t o r ) Γ < # ( S ( B / K ) / ( κ ) ) Γ < (*) # X B / K ∞ t o r Γ < ∞ ⇔ # S B / K ∞ / ( κ ) Γ < ∞ {:(*)#(X(B//K_(oo))_(tor))_(Gamma) < oo<=>#(S(B//K_(oo))//(kappa))_(Gamma) < oo:}\begin{equation*} \#\left(X\left(B / K_{\infty}\right)_{\mathrm{tor}}\right)_{\Gamma}<\infty \Leftrightarrow \#\left(S\left(B / K_{\infty}\right) /(\kappa)\right)_{\Gamma}<\infty \tag{*} \end{equation*}(*)#(X(B/K∞)tor)Γ<∞⇔#(S(B/K∞)/(κ))Γ<∞
Now corank Z p Sel p ( A / Q ) = 1 Z p Sel p ∞ ⁡ ( A / Q ) = 1 Z_(p)Sel_(p oo)(A//Q)=1\mathbb{Z}_{p} \operatorname{Sel}_{p \infty}(A / \mathbb{Q})=1ZpSelp∞⁡(A/Q)=1 implies that the left-hand side of ( ) ( ∗ ) (**)(*)(∗) holds. On the other hand, under the Gross-Zagier formula, ord s = 1 L ( s , A / Q ) = 1 s = 1 L ( s , A / Q ) = 1 _(s=1)L(s,A//Q)=1{ }_{s=1} L(s, A / \mathbb{Q})=1s=1L(s,A/Q)=1 is a consequence of the righthand side of ( ) ( ∗ ) (**)(*)(∗).
First proof of HPMC. The two variable Rankin-Selberg p p ppp-adic L-function L p ( f × χ ) L p ( f × χ ) L_(p)(f xx chi)\mathscr{L}_{p}(f \times \chi)Lp(f×χ) (see [21]) associated to ( f , χ ) ( f , χ ) (f,chi)(f, \chi)(f,χ) has a decomposition in terms of L p ( λ ) L p ( λ ) L_(p)(lambda)\mathscr{L}_{p}(\lambda)Lp(λ) and L p ( λ / χ χ ) L p λ ∗ / χ ∗ â‹… χ L_(p)(lambda^(**)//chi^(**)*chi)\mathscr{L}_{p}\left(\lambda^{*} / \chi^{*} \cdot \chi\right)Lp(λ∗/χ∗⋅χ), where L p ( λ ) , L p ( λ / χ χ ) L p ( λ ) , L p λ ∗ / χ ∗ â‹… χ L_(p)(lambda),L_(p)(lambda^(**)//chi^(**)*chi)\mathscr{L}_{p}(\lambda), \mathscr{L}_{p}\left(\lambda^{*} / \chi^{*} \cdot \chi\right)Lp(λ),Lp(λ∗/χ∗⋅χ) are the Katz p p ppp-adic L-functions (see [29]) associated to λ λ lambda\lambdaλ and λ / χ λ ∗ / χ ∗ lambda^(**)//chi^(**)\lambda^{*} / \chi^{*}λ∗/χ∗. χ χ chi\chiχ, respectively. Note that L p ( f × χ ) L p ( f × χ ) L_(p)(f xx chi)\mathscr{L}_{p}(f \times \chi)Lp(f×χ) and L p ( λ ) L p ( λ ) L_(p)(lambda)\mathscr{L}_{p}(\lambda)Lp(λ) vanish along the anticyclotomic line, thus we may consider their derivatives with respect to the cyclotomic variable, i.e.,
( L p ( f × χ ) ) = ( L p ( λ ) ) ( L p ( λ / χ χ ) ) L p ′ ( f × χ ) = L p ′ ( λ ) L p λ ∗ / χ ∗ â‹… χ (L_(p)^(')(f xx chi))=(L_(p)^(')(lambda))(L_(p)(lambda^(**)//chi^(**)*chi))\left(\mathscr{L}_{p}^{\prime}(f \times \chi)\right)=\left(\mathscr{L}_{p}^{\prime}(\lambda)\right)\left(\mathscr{L}_{p}\left(\lambda^{*} / \chi^{*} \cdot \chi\right)\right)(Lp′(f×χ))=(Lp′(λ))(Lp(λ∗/χ∗⋅χ))
The HPMC is based on Λ Î› Lambda\LambdaΛ-adic Gross-Zagier formula and Rubin's main conjecture.
On the one hand, the Λ Î› Lambda\LambdaΛ-adic Gross-Zagier formula [21] connects Heegner point with L p ( f × χ ) L p ′ ( f × χ ) L_(p)^(')(f xx chi)\mathscr{L}_{p}^{\prime}(f \times \chi)Lp′(f×χ) as
( L p ( f × χ ) ) = ( κ , κ ) = Char ( S ( B / K ) / ( κ ) ) R ( f × χ ) L p ′ ( f × χ ) = ( ⟨ κ , κ ⟩ ) = Char ⁡ S B / K ∞ / ( κ ) R ( f × χ ) (L_(p)^(')(f xx chi))=((:kappa,kappa:))=Char(S(B//K_(oo))//(kappa))R(f xx chi)\left(\mathscr{L}_{p}^{\prime}(f \times \chi)\right)=(\langle\kappa, \kappa\rangle)=\operatorname{Char}\left(S\left(B / K_{\infty}\right) /(\kappa)\right) R(f \times \chi)(Lp′(f×χ))=(⟨κ,κ⟩)=Char⁡(S(B/K∞)/(κ))R(f×χ)
where , ⟨ â‹… , â‹… ⟩ (:*,*:)\langle\cdot, \cdot\rangle⟨⋅,⋅⟩ is the Λ Î› Lambda\LambdaΛ-adic height pairing and R ( f × χ ) R ( f × χ ) R(f xx chi)R(f \times \chi)R(f×χ) is the Λ Î› Lambda\LambdaΛ-adic regulator of f × χ f × χ f xx chif \times \chif×χ which is nonzero by the rigidity principle [5]. On the other hand, Rubin's main conjecture [ 1 , 43 ] [ 1 , 43 ] [1,43][1,43][1,43] implies
( L p ( λ ) ) = Char ( X ( λ ) tor ) R ( λ ) , ( L p ( λ / χ χ ) ) = Char ( X ( λ / χ χ ) ) L p ′ ( λ ) = Char ⁡ X ( λ ) tor R ( λ ) , L p λ ∗ / χ ∗ â‹… χ = Char ⁡ X λ ∗ / χ ∗ â‹… χ (L_(p)^(')(lambda))=Char(X(lambda)_(tor))R(lambda),quad(L_(p)(lambda^(**)//chi^(**)*chi))=Char(X(lambda^(**)//chi^(**)*chi))\left(\mathscr{L}_{p}^{\prime}(\lambda)\right)=\operatorname{Char}\left(X(\lambda)_{\operatorname{tor}}\right) R(\lambda), \quad\left(\mathscr{L}_{p}\left(\lambda^{*} / \chi^{*} \cdot \chi\right)\right)=\operatorname{Char}\left(X\left(\lambda^{*} / \chi^{*} \cdot \chi\right)\right)(Lp′(λ))=Char⁡(X(λ)tor)R(λ),(Lp(λ∗/χ∗⋅χ))=Char⁡(X(λ∗/χ∗⋅χ))
where R ( λ ) R ( λ ) R(lambda)R(\lambda)R(λ) is the Λ Î› Lambda\LambdaΛ-adic regulator which is nonzero [5], X ( λ ) X ( λ ) X(lambda)X(\lambda)X(λ) and X ( λ / χ χ ) X λ ∗ / χ ∗ â‹… χ X(lambda^(**)//chi^(**)*chi)X\left(\lambda^{*} / \chi^{*} \cdot \chi\right)X(λ∗/χ∗⋅χ) are certain anticyclotomic Selmer groups. Then, the HPMC follows from the decomposition
Char ( X ( B / K ) tor ) = Char ( X ( λ ) tor ) Char ( X ( λ / χ χ ) ) Char ⁡ X B / K ∞ tor = Char ⁡ X ( λ ) tor Char ⁡ X λ ∗ / χ ∗ â‹… χ Char(X(B//K_(oo))_(tor))=Char(X(lambda)_(tor))Char(X(lambda^(**)//chi^(**)*chi))\operatorname{Char}\left(X\left(B / K_{\infty}\right)_{\operatorname{tor}}\right)=\operatorname{Char}\left(X(\lambda)_{\operatorname{tor}}\right) \operatorname{Char}\left(X\left(\lambda^{*} / \chi^{*} \cdot \chi\right)\right)Char⁡(X(B/K∞)tor)=Char⁡(X(λ)tor)Char⁡(X(λ∗/χ∗⋅χ))
and the comparison of Λ Î› Lambda\LambdaΛ-adic regulators, R ( λ ) = R ( f × χ ) R ( λ ) = R ( f × χ ) R(lambda)=R(f xx chi)R(\lambda)=R(f \times \chi)R(λ)=R(f×χ).
Second proof of HPMC. Via Λ Î› Lambda\LambdaΛ-adic Waldspurger formula and nontriviality of κ κ kappa\kappaκ, the HPMC is equivalent to the BDP main conjecture [45]. Let p = v v ¯ p = v v ¯ p=v bar(v)p=v \bar{v}p=vv¯ where v v vvv is determined by ι p ι p iota_(p)\iota_{p}ιp.
Proposition 34. The Λ Q p Λ Q p Lambda_(Q_(p))\Lambda_{\mathbb{Q}_{p}}ΛQp-modules
S v ( B / K ) := ( lim n lim m m Sel p m , v ( B / K n ) ) Q p , X v ( B / K ) := ( lim n lim m Sel p m , v ( B / K n ) ) Q p S v B / K ∞ := lim n   lim m   m Sel p m , v ⁡ B / K n Q p , X v B / K ∞ := ( lim n lim m   Sel p m , v ⁡ B / K n Q p ∨ {:[S_(v)(B//K_(oo)):=(lim _(n)lim_(m)_(m)Sel_(p^(m),v)(B//K_(n)))_(Q_(p))","],[X_(v)(B//K_(oo)):=(lim_(n)lim _(m)Sel_(p^(m),v)(B//K_(n)))_(Q_(p))^(vv)]:}\begin{aligned} & S_{v}\left(B / K_{\infty}\right):=\left(\underset{n}{\lim } \underset{m}{\lim _{m}} \operatorname{Sel}_{\mathfrak{p}^{m}, v}\left(B / K_{n}\right)\right)_{\mathbb{Q}_{p}}, \\ & \left.X_{v}\left(B / K_{\infty}\right):=\underset{n}{(\lim } \underset{m}{\lim } \operatorname{Sel}_{\mathfrak{p}^{m}, v}\left(B / K_{n}\right)\right)_{\mathbb{Q}_{p}}^{\vee} \end{aligned}Sv(B/K∞):=(limnlimmmSelpm,v⁡(B/Kn))Qp,Xv(B/K∞):=(limnlimmSelpm,v⁡(B/Kn))Qp∨
are finitely generated torsion Λ Q p Λ Q p Lambda_(Q_(p))\Lambda_{\mathbb{Q}_{p}}ΛQp-modules. Here Sel p m , v Sel p m , v Sel_(p^(m),v)\operatorname{Sel}_{p^{m}, v}Selpm,v is the p m p m p^(m)\mathfrak{p}^{m}pm-Selmer group with v v vvv relaxed and v ¯ v ¯ bar(v)\bar{v}v¯-strict local Selmer condition [45].
Let L v ( B / K ) L v B / K ∞ L_(v)(B//K_(oo))\mathscr{L}_{v}\left(B / K_{\infty}\right)Lv(B/K∞) be the anticyclotomic BDP p p ppp-adic L-function in [3,38].
Conjecture 35 (BDP Main Conjecture). Char ( X v ( B / K ) ) = ( L v ( B / K ) ) X v B / K ∞ = L v B / K ∞ (X_(v)(B//K_(oo)))=(L_(v)(B//K_(oo)))\left(X_{v}\left(B / K_{\infty}\right)\right)=\left(\mathscr{L}_{v}\left(B / K_{\infty}\right)\right)(Xv(B/K∞))=(Lv(B/K∞)).
The Λ Q p Λ Q p Lambda_(Q_(p))\Lambda_{\mathbb{Q}_{p}}ΛQp-modules S v ( B / K ) , X v ( B / K ) , ( L v ( B / K ) ) S v B / K ∞ , X v B / K ∞ , L v B / K ∞ S_(v)(B//K_(oo)),X_(v)(B//K_(oo)),(L_(v)(B//K_(oo)))S_{v}\left(B / K_{\infty}\right), X_{v}\left(B / K_{\infty}\right),\left(\mathscr{L}_{v}\left(B / K_{\infty}\right)\right)Sv(B/K∞),Xv(B/K∞),(Lv(B/K∞)) can be decomposed in terms of Selmer groups and p p ppp-adic L-functions of λ , λ / χ χ λ , λ ∗ / χ ∗ â‹… χ lambda,lambda^(**)//chi^(**)*chi\lambda, \lambda^{*} / \chi^{*} \cdot \chiλ,λ∗/χ∗⋅χ. Then, we approach the BDP main conjecture based on Iwasawa main conjecture for imaginary quadratic fields proved by Rubin [43].
The second approach generalizes to CM elliptic curves over totally real field [ 6 , 30 ] [ 6 , 30 ] [6,30][6,30][6,30].

ACKNOWLEDGMENTS

The author thanks Ashay Burungale, John Coates, Henri Darmon, Yifeng Liu, Christopher Skinner, Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang for their helpful comments, discussions, and constant support.

FUNDING

This work was partially supported by NSFC grant #11688101.

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YE TIAN

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China, ytian@math.ac.cn

ARITHMETIC AND GEOMETRIC LANGLANDS PROGRAM

XINWEN ZHU

ABSTRACT

We explain how the geometric Langlands program inspires some recent new prospectives of classical arithmetic Langlands program and leads to the solutions of some problems in arithmetic geometry.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 11R39; Secondary 11S37, 14D24, 11G18, 11G40

KEYWORDS

Classical Langlands correspondence, geometric Langlands program, Shimura varieties
The classical Langlands program, originated by Langlands in 1960s [41], systematically studies reciprocity laws in the framework of representation theory. Very roughly speaking, it predicts the following deep relations between number theory and representation theory:
A special case of this correspondence, known as the Shimura-Tanniyama-Weil conjecture, implies Fermat's last theorem (see [62]).
The geometric Langlands program [42], initiated by Drinfeld and Laumon, arose as a generalization of Drinfeld's approach [20] to the global Langlands correspondence for G L 2 G L 2 GL_(2)\mathrm{GL}_{2}GL2 over function fields. In the geometric theory, the fundamental object to study shifts from the space of automorphic forms of a reductive group G G GGG to the category of sheaves on the modul space of G G GGG-bundles on an algebraic curve.
For a long time, developments of the geometric Langlands were inspired by problems and techniques from the classical Langlands, with another important source of inspiration from quantum physics. The basic philosophy is known as categorification/geometrization. In recent years, however, the geometric theory has found fruitful applications to the classical Langlands program and some related arithmetic problems. Traditionally, one applies Grothendieck's sheaf-to-function dictionary to "decategorify" sheaves studied in geometric theory to obtain functions studied in arithmetic theory. This was used in Drinfeld's approach to the Langlands correspondence for G L 2 G L 2 GL_(2)\mathrm{GL}_{2}GL2, as mentioned above. Another celebrated example is Ngô's proof of the fundamental lemma [55]. In recent years, there appears another passage from the geometric theory to the arithmetic theory, again via a trace construction, but is of different nature and is closely related to ideas from physics. V. Lafforgue's work on the global Langlands correspondence over function fields [39] essentially (but implicitly) used this idea.
In this survey article, we review (a small fraction of) the developments of the geometric Langlands program, and discuss some recent new prospectives of the classical Langlands inspired by the geometric theory, which in turn lead solutions of some concrete arithmetic problems. The following diagram can be regarded as a road map:
Notations. We use the following notations throughout this article. For a field F F FFF, let Γ F ~ / F Γ F ~ / F Gamma_( tilde(F)//F)\Gamma_{\tilde{F} / F}ΓF~/F be the Galois group of a Galois extension F ~ / F F ~ / F tilde(F)//F\tilde{F} / FF~/F. Write Γ F = Γ F ¯ / F Γ F = Γ F ¯ / F Gamma_(F)=Gamma_( bar(F)//F)\Gamma_{F}=\Gamma_{\bar{F} / F}ΓF=ΓF¯/F, where F ¯ F ¯ bar(F)\bar{F}F¯ is a separable closure of F F FFF. Often in the article F F FFF will be either a local or a global field. In this case, let W F W F W_(F)W_{F}WF denote the Weil group of F F FFF. Let cycl denote the cyclotomic character.
For a group A A AAA of multiplicative type over a field F F FFF, let X ( A ) = Hom ( A F ¯ , G m ) X ∙ ( A ) = Hom ⁡ A F ¯ , G m X^(∙)(A)=Hom(A_( bar(F)),G_(m))\mathbb{X}^{\bullet}(A)=\operatorname{Hom}\left(A_{\bar{F}}, \mathbb{G}_{m}\right)X∙(A)=Hom⁡(AF¯,Gm) denote the group of its characters, and X ( A ) = Hom ( G m , A F ¯ ) X ∙ ( A ) = Hom ⁡ G m , A